Largest bipartite subgraphs in triangle-free graphs with maximum degree three

Largest Bipartite Subgraphs in Triangle= Free Graphs with Maximum Degree Three

J.A. Bondy UNIVERSITY OF WATERLOO

WATERLOO, ON TARIO CANADA N2L 3G7

S . C . Locke FLORIDA ATLANTIC UNIVERSITY

BOCA RATON, FLORIDA 33437, USA

ABSTRACT

Let G be a triangle-free, loopless graph with maximum degree three. We display a polynomi$ algorithm which returns a bipartite subgraph of G containing at least 5 of the edges of G. Furthermore, we characterize the dodecahedron and the Petersen graph as the only 3-regular, triangle-free, loopless, connected graphs for which no bipartite subgraph has more than this proportion.

1. INTRODUCTION

It was proved by Karp [12], Even and Shiloach [7], and Garey, Johnson, and Stockmeyer [S] that the problem of finding a bipartite subgraph of a given graph with the maximum number of edges is NP-complete. Thus, the efficient determination of “large” bipartite subgraphs is a question of both practical and theoretical interest.

To eliminate trivialities, we restrict our attention to connected loopless graphs with at least two vertices. Our notation and terminology is that of Bondy and Murty 1 1 1 . For a graph G , the set of vertices is denoted by V ( G ) , the set of edges of E ( G ) and the cardinalities of these sets by v ( G ) and E ( G ) , respectively. The degree of the vertex u in G is written d,(u). If all vertices have degree r, we say that G is r-regular. N,(u) represents the set of neighbors of u in G . If X is a set of vertices, N G ( X ) is the union of the neighbor sets of the ver-

Journal of Graph Theory, Vol. 10 (1986) 477-504 C 1986 by John Wiley & Sons, Inc. CCC 0364-90241861040477-28$04.00

478 JOURNAL OF GRAPH THEORY

tices in X . When there is no ambiguity regarding which graph is being considered, we shall omit the G from these symbols. For two sets X and Y , the symmetric difference of X and Y is denoted by X A Y . K , is the complete graph on r vertices. A graph is trianglefree if it has no subgraph isomorphic to Ks. For a set of vertices 2, G[Z] is the subgraph of G induced by Z and G - Z is

Let G be a graph. Let B be any bipartite subgraph of G with the maximum G[V(G) - 21.

number of edges, and let

We shall refer to h(G) as the bipartite density of G.

tite subgraph B with Erdos [4,5] and Edwards [3] showed that, for any graph G , there is a bipar-

E(B) 2 fe(G) + ~ ( u ( G ) - 1 ) . (1.1)

Erdos and Lovasz (see Erdos [6]) provcd, moreover, that the lower bound on b(G) given in (1.1) cannot be improved substantially by requiring G to have large girth. It can be improved, however, by imposing certain other constraints. For instance, Erdos [6] proved that a 2m-colorable graph satisfies

If G is r-regular, where r 2 3, and if G # KrTI , then it follows from Brooks' theorem [2] that G is r-colorable. Hence, (1.2) gives a lower bound on thc bipartite density of a regular graph. For 3-regular graphs, one obtains

Staton [20] and Locke [ 14, 151 improved (1.3) by showing that, for 3-regular graphs other than K4,

Equality holds in (1.4) for an infinite number of connected 3-regular graphs. Bondy and Locke (see Locke [ 161) display a relationship between the bipartite density and the independence number in 3-regular graphs.

LARGEST BIPARTITE SUBGRAPHS 479

Hadlock [lo] proved that b(G) can be calculated in polynomial time for a planar graph G . Grijtschel and Pulleyblank [9] described a class of graphs (including all planar graphs) for which a linear program will compute b(G) in polynomial time. However, Yannakakis 1211 showed that the problem of determining the bipartite density of a graph remains NP-complete even if G is 3-regular and triangle-free. Hopkins and Staton [ I l l proved that, if G is 3-regular and triangle-free, then

b(G) 2 :. (1.5)

We extend the work of Hopkins and Staton by displaying a polynomial algorithm which, given a triangle-free graph with maximum degree three, finds a bipartite subgraph B' with

In Section 7, we extend a result of Mallc [ 171 by showing that the only triangle- free 3-regular graphs for which equality holds in (1.5) are the Petersen graph and the dodecahedron. (Robertson [ 191 claims that this result can be derived from [ 181.) We then deduce that the only 3-regular graph which contain a set of 5-cycles covering every edge of G exactly m times, for some m, are these same two graphs.

2. SWITCHING AND COVERING

To obtain lower bounds on b(G), we shall begin with an arbitrary bipartite subgraph B with bipartition ( X , Y ) , and consider the effects of local changes in X and Y on the number of edges in B . Since we are considering maximum bipartite subgraphs, we shall restrict our attention to the case in which B is the cut [ X , Y 1.

For a set of vertices 2, we shall denote by BAZ the bipartite subgraph of G with bipartition ( X A Z , Y A Z ) and with all edges which have one end in XAZ and the other in Y A Z . This operation can be thought of as switching the colors of the vertices of Z . Its effect is to replace the edges of the cut [Z, V(G) - Z ] which are in B by those which are not in B . Clearly, any maximum bipartite subgraph can be obtained from an arbitrary bipartite subgraph by switching an appropriate set of vertices. Thus, we have the following lemma.

Lemma 2.1. Let G be a graph and let B be a bipartite subgraph of G. Then either B is maximum or there is a bond C in G such that B contains fewer than one-half of the edges of C .

Proof. Clearly, if there is such a bond C, switching the colors on one side of C results in a bipartite subgraph with a larger number of edges.

Hence, we need only consider the case in which B is not maximum. Let B ' be a maximum bipartite subgraph. Then C' = E(B)AE(B') is a cut in G such that B contains fewer than one half of the edges of C' . But C' decomposes into bonds, at least one of which satisfies the conclusion of the lemma. I

We shall be interested in the effects of switching small sets of vertices. Whcn we restrict ourselves to switching just one vertex at a time, we obtain a result of Erdos [5] which we state without proof.

Lemma 2.2. Then, for every vertex u of G,

Let G be a graph and let B be a largest bipartite subgraph of G.

Corollary 2.3 is an immediate consequence of Lemma 2.2.

Corollary 2.3. Let G be a graph. Then

An example is given in [14] demonstrating that a larger constant than that

We obtain upper bounds on b(G) by using the following lemma, whose proof given in Corollary 2.3 cannot be obtained by switching pairs of vertices.

may be found in [ 141.

Lemma 2.4. that each edge of G is contained in at least rn of the subgraphs H i . Then

Let G be a graph and let H , , H 2 , . . . , H k be subgraphs of G such

If every edge of G is contained in exactly rn of the subgraphs H , , i = 1,2, . . . , k , and if each H i is isomorphic to the same graph H , we shall call {Hi }%, an m-covering ofC by H . In this case, we note that k e ( H ) = rns(G), and hence, b(G) I b ( H ) . In particular, we shall be interested in the case when H is a 5-cycle.

Corollary 2.5. Let G be a graph which has an rn-covering by 5-cycles. Then

The dodecahedron is 2-covered by its 5-cycles; the Petersen graph is 4-covered by its 5-cycles. Hence the bipartite density of each of these two graphs is at most $ and thus equal to $.

3. TRIANGLE-FREE GRAPHS WITH MAXIMUM DEGREE THREE

We shall henceforth restrict our attention to triangle-free graphs with maximum degree three. Some cases are most easily handled recursively.

Lemma 3.1. Let G be a triangle-free graph with maximum degree at most three. Further, suppose that b(G') 2 $ for all such graphs G ' with fewer vertices than G. Then b(G) 2 $ if G has either of the following two properties:

(i) G is not 2-connected; or (ii) d(x) = d( y) = 2, for some pair of vertices x , y with distance at most two.

Moreover, if G is connected, but not 2-connected, then b(G) > 4 .

Proof. (i) If G is connected, but not 2-connected, then G has a cut edge e , because the maximum degree is at most three. Let the components of G - e be GI and G2. Then

Thus b(G) > $. If C is disconnected, with components G,, 1 5 i 5 rn, then

2 $E &(Gi) i- 1

= $&(G) .

Thus b(G) 2 $, with equality if and only if

&(Gi) > 0 3 b(Gi) = $, 1 5 i 5 rn

(ii) We may now assume that G is 2-connected. Suppose first that d(x,p) = I . Then either G is a cycle or x and y lie on a path P, whose internal vertices have degree two and whose termini have degree three. If G is a cycle, then b(G) 2 :, with equality if and only if G is a pentagon. Thus we may assume that x , y have neighbors w and z , respectively, where d(z) = 3. Let r and s be the other neighbors of z. We shall construct a bipartite subgraph B having at least 4 of the edges of G.

Let B ' be a maximum bipartite subgraph of G - {x, y, z}. If r and s are in the same color class of B ' , let B contain B ' U {rz, sz,yz,xy} and possibly wx. If r and s have different colors in B ' , let B contain B ' U {wx,xy,yz} and either zr or zs (see Figure 3.1(a)). In either case,

&(B) 1 & ( B ' ) + 4

2 $ ( E ( G ) - 5 ) + 4

= ~ E ( G ) .

Thus b ( G ) 2 $, with equality if and only if h(G - {x,?, z } ) = 4 and w x @

E ( B ) in the first case. It remains only to consider the case in which d(x , y ) = 2. We may suppose

that x and y are vertices of degree two with a common neighbor z , where &(z) = 3. Let r , s, and r be the other neighbors of x, y , and z , respectively, and let B ' be a maximum bipartite subgraph in G - {x, y, z} . If r and s are in the same color class of B ' , then let B contain B ' U ( rx ,xz , zy ,ys} and possibly zt. If r and s are in different color classes of B ', then let B contain B ' U { z f , zy , zx} and one of rx or ys (see Figure 3.1(b)). In either case,

&(B) 5 & ( B ' ) + 4

5 $(E(G) - 5 ) + 4

= $ & ( G ) .

Thus b(G) 2 $, with equality if and only if b(G - { x , y , z}) = $ and zt e E(B) in the first case.

This completes the proof of Lemma 3.1. I

We also note that a minimum counterexample to ( I .5) cannot have multiple edges. If there is a pair of edges between some two vertices x and y, then we proceed in the same manner as we did for a pair of adjacent vertices of degree two in the proof of Lemma 3.1.

FIG. 3.1

We shall thus restrict our attention to the class of triangle-free graphs with maximum degree three which do not satisfy either condition (i) or condition (ii) in the statement of Lemma 3 . 1 . By (9 we denote the class of simple triangle- free graphs G with maximum degree three which satisfy the conditions:

(i) G is 2-connected; and (ii) Thzre are no vertices x , y i n G with d G ( x ) = d , ( y ) = 2 and with

d(x ,y ) I 2 .

4. WEIGHT FUNCTION

D,i = {u E V(G): dB(u) = i and &(u) = j } ,

and let dij = IDi,/. By Lemma 2.2 and our assumption that G is 2-connected, the only nonempty sets d i j are D 1 2 , D22r 0 2 3 , and D33. Therefore, the bipartite density of G is given by:

To prove that b(G) 1 :, we must therefore prove that

or, equivalently, that

In order to verify (4. l), we first assign, to each vertex u, an integral weight w(u) as follows:

3 if u E D 3 3 , 2 if u E D22r

- 2 if u E Dz3, -3 i f u E D I 2 .

w(u) =

For a subset X of V(G), we let

w(X) = c w ( u ) , UEX

Now, (4.1) is equivalent to

w(V(G) ) 2 0 . (4.2)

We shall prove (4.2) by partitioning V ( G ) into subsets, each of which has non- negative weight. To do this, the structure of the subgraph B must be examined in some detail.

5. FORBIDDEN SUBGRAPHS

We have already seen that certain subgraphs are forbidden in B , for instance, vertices in DI3. To prove (4.2), we now consider various other forbidden subgraphs.

In the subgraph B , the vertices of degree two lie on cycles or on paths whose ends are in DI2 U D33, For convenience, we shall refer to these paths as D-paths. More precisely, we shall call any path which has both termini in DI2 U D3, and all internal vertices in DZ2 U D2, a D-path.

Lemma 5.1. Let G be a graph in (3, and let B be a maximum bipartite subgraph of G. Let P = wowl . * . w, be a path in B with wi € DZ2 U DZ3, 0 I i 5 m. Then P has at most two vertices in D23.

Proof. Suppose w,, w,, wk E DZ3, for i < j < k, with no other vertices of DZ3 separating w, , w,, and wk on P. If there is no edge in E ( G ) - E(B) con- necting any pair of these thrcc vertices, then

&(BA{wi , w , , I , . . . , wk} ) = E(B) + 1 ,

Contradicting thc maximality of R . If w,w, E E(G) - E(B) , where r < s , and r , 5 E {i, j , k}, then I' = dmod 2) since B is maximum. Also, s # r + 2 since G has no triangles. Thus s 2 r + 4, implying that at least two of w , + ~ , w r r 2 , and w,+~ have degree two. Hence G violates condition (ii) in the definition of 3, contradicting the hypothesis that G E (3. This completes the proof of Lemma 5.1. I

We can now identify all D-paths. Let P = wowl wk be a D-path. The ver-

Suppose first that one of these vertices, say w t , is in DI2 . Since G E 3, tices wo and wk may be in DI2 or D33.

& ( w ~ - ~ ) = 3. Thus wkel € D23 or wk-l E Dj3. If wk-l E 0 2 3 then

contradicting the maximality of B. Thus w ~ - ~ E D33 (and k = 1).

since G E 3, at most two vertices of D22. If wo and wk are both in D33, then P contains at most two vertices of 0 2 3 and,

Figure 5.1 displays all possible D-paths. In these diagrams, as well as in all succeeding diagrams, we represent vertices of degree three in B by squares, and vertices of degree one or two in B by circles. Broken lines represent edges of E(G) - E ( B ) .

We also note that B[DZ2 U 0 2 3 1 has no paths of length greater than three and, hence, no cycles.

To each D-path P and each endvertex x of P which is in D33, we assign an integer w(P, x) which we shall call the projection of P onto x. The projections of each D-path onto its termini u and u are given in Figure 5.1. Note that w(P, u) + w(P, u) = w(V(P) - D33) if both u and u are in D33, and w(P, u) = w(V(P) - D34 if u E DI2. If P is the edge uu, with u, u E D33, we shall call P a strong edge.

We shall now consider how these D-paths meet vertices in Dj3.

Lemma 5.2. w(P,x) 5 w(Q,x) 5 w(R,x). Then exactly one of the following holds:

Let x E D33, and let the D-paths leaving .r be P, Q, and R with

(i) w(P,x) = -3, w(Q,x) = -1, and R is a strong edge; (ii) w(P,x) = -3 and each of Q and R is a strong edge; (iii) w(P,x) = w(Q,x) = -2, R is a strong edge, and there is an edge with

one end on P and the other end on Q (as in Figure 5.2); (iv) w(P,x) = w(Q,x) = -2 and w(R,x) = 1; (v) w(P,x) = -2, w(Q,x) 2 -1, and w(R,x) 2 0; (vi) w(P,x) 2 -1.

Proof. Let a, c , and L' be the neighbors of x with P beginning x u , Q begin-

Suppose first w(P,x) = -3. Then a E D I 2 . Since G E %, d G ( c ) = ning xc, and R beginning xe.

dG(e) = 3. If c and e are in 023, then since G is triangle-free,

&(BA(u, c, e , x } ) 2 E(B) + 1 ,

contradicting the maximality of B. Thus, without loss of generality, e E Dj3 and therefore R = xe is a strong edge.

FIG. 5.1

\ FIG. 5.2

If w(Q,x) = -2 and Q begins with c , d E DZ3, then

~ ( B A { u , c , d , x } ) 2 E(B) + 1 ,

again Contradicting the maximality of B. Therefore w(Q, x) 2 - 1 . But, since G E %, w(Q,x) 5 0. Thus w(Q,x) = - 1 or w(Q,x) = 0. These two possibilities are listed as cases (i) and (ii) in the statement of thc lemma.

Suppose now w(P,x) = -2 and w(Q,x) = -2. Let P begin xub and Q begin xcd. Since B is maximum,

&(BA{u, b, c , d, x}) 5 E ( B ) .

Thus, there must be an edge of E ( G ) - E ( B ) in G [ { u , b , c , d } J . Since G is triangle-free, this is the edge bd (see Figure 5.2). If e E D33, then the structure is that given in Case (iii). If e E D I 2 or e E Dz3 then

&(BA{a, b, c, e,x}) 2 E(B) + 1 ,

contradicting the choicc of B. Thus e E D2:. Let e have the neighbors x andf in B . IfJ’ E LA3, then

again contradicting the choice of B . Thusf E D33 and the situation is as given in Case (iv).

Suppose now w(P, x) = -2 and w(Q, x) = - 1. If e E D , 2 or e E D23 then

c(BA{a, b, c , e,x}) 2 (B) + I .

Thus e E DZ2 or e E D 3 3 , therefore w(R,x) 2 0 and the situation is that described in Case (v).

The remaining case is w(P, x) 2 - 1, which is Case (vi). I

6. REVISED WEIGHT FUNCTION

For x E D33, with D-paths P , Q , and R leaving x , define w’(x) = w(x) + w ( P , x ) + w ( Q , x ) + w(R,x ) . The revised weights of the vertex x in the six cases of Lemma 5.2 are:

(i) wr(.r) = -1; (ii) w’(x) = 0; (iii) w’(x) = - 1; (iv) w ’ ( x ) 2 0; (v) wr(x) 2 0; and (vi) w’(x) 2 0 .

Thus in all cases w ’ ( x ) 2 - 1. For a subset S of D33, define

w ’ ( S ) = c w ’ ( x ) . X E S

Then (4.2) is equivalent to

We shall prove (6.1) by considering certain subsets of D33 with at most four vertices. More precisely, we consider a spanning forest F of G\D33], and partition the vertices of each component of F into sets of size at most four.

Lemma 6.1. Let x E D33, with no neighbor of x in D33. Then w’(x) 2 0.

Proof’. The only possibilities for the D-paths leaving x are Cases (iv), (v), and (vi) in the statement of Lemma 5.2. In each case w’(x) 2 0. I

Lemma 6.2. Let xy E E(B[D,,]), with w ‘ ( x ) = - 1. Then w’(y) 2 1. Fur- thermore, if yuu is a path in B with u, u € D2,, then u is adjacent, by an edge in E(G) - E(B) , to a neighbor of x other than y.

Proof’. Since w ’(x) = - 1 , x is of type (i) or (iii) in the statement of Lemma 5 . 2 . Suppose x is of type (i) with neighbors a and b. If y is adjacent to c, d E D23, then

contradicting the choice of B . If y is adjacent to c E DI2, then

&(BA{a, 6, c , x , y } ) 2 c(B) + 1 ,

also a contradiction. Thus y is adjacent to no vertices in D12 and at most one vertex in D23. Therefore w ’ ( y ) 2 1 and w({x,y}) 2 0. Suppose P is a D-path beginning yuu, with w(P,y) = -2. If ua G! E(G) - E ( B ) and ub G! E ( G ) - E ( B ) (we also admit the case b = u) , then

c(BA{a, 6 , u, u , x , y}) 2 c(B) + 1 .

Therefore, ua E E(G) - E(B) or uh E E(G) - E(B). Thus we may assume that x is of type (iii), with D-paths P beginning xu6

and Q beginning xcd as in Figure 5 . 2 . If y is adjacent to, say h, (and possibly also d ) then

c(BA{a, b , c , x , y } ) 2 E(B) + 1 .

If y is adjacent to e E DI2, then

&(BA{a, 6, c, e , x , y } ) 2 E(B) + 1

If y is adjacent to e,f E 0 2 3 , then

Thus y is adjacent to no vertices in D I 2 and at most one vertex in DZ3. Again, w’( {x , y } ) 2 0. Suppose P ’ is a D-path beginning yuu with w(P’ ,y ) = - 2 . If ua e E(G) - E(B) and uc e E(G) - E(B) , then

r(BA{a, h, c , u , u , x , y } ) 2 E(B) + 1 .

Therefore, ua E E ( G ) - E(B) or uc E E ( G ) - E(B). I

Lemma 6.3. Let xyz be a path in BLD,,]. Then w ’(x) + w ‘( y ) + w ’ ( z ) 2 0.

Proof. By Lemma 6 . 2 , w’ (x ) + w ’ ( y ) 2 0 and w ’ ( y ) + w‘(z) 2 0. Thus w’(x ) + w ’ ( y ) + w’(z) 2 - w ’ ( y ) . If ~ ’ ( y ) 5 0 , we arc done. Therefore we may assume that w ‘ ( y ) 2 1 . Then the only possible case is w ’ ( x ) = - 1 , w ’ ( y ) = 1 and w ’ ( z ) = - 1 . In this case, x and z are of type ( i ) or (iii) in Lemma 5 . 2 . Since w ‘ ( y ) = I , there is a path y u u with u , u E D 2 3 . By Lemma 6 . 2 , an edge of E ( G ) - E(B) joins u to a neighbor of x (but not y ) and to a neighbor of z (but not y ) . There is only one edge of E ( G ) - E(B) incident with u. Thus there is a vertex w E ( N ( x ) n N ( z ) ) - { y } . Therefore x and z can only be of type (i) in Lemma 5 . 2 . Let x ‘ and z’ be thc neighbors of x and z in DI2 . Since G € 3, x’z’ e E(G) . Thus

Therefore w ’({x, y, z } ) r 0. I

Lemma 6.4. Let wx, wy, wz E E ( B [ D 3 J ) . Then w’({w,x,y ,z}) 2 0.

Proof. Since w ‘ ( w ) = 3 and each of w ’ ( x ) , w ’ ( y ) , and ~ ‘ ( z ) is at least - 1, this result is immediate. I

Theorem 6.5. Then b(G) 2 i .

Let G be a triangle-free graph with maximum degree three.

Proof. Let H I , i = 1,2 , . . . , m , be the connected components of B[D33J. To prove Theorem 6.5, we need only prove that w‘(V(H,)) 2 0, for each i, i = 1,2 , . . . , m. Then ~ ’ ( 0 ~ ~ ) 2 0 and, by (6. l ) , (4.2), and (4.1), b(G) 2 2.

Let TI be a spanning tree of H I . We shall prove that w’(V(T,)) 2 0 by induction on the number of vertices of subtrees T of TI. If v ( T ) 5 3 or if T K 1 . 3 , then w’(V(T)) I 0 by one of Lemmas 6.1 through 6.4. Thus we may assume that v ( T ) 2 4 and T $ K,,3. Let x be a vertex of T furthest from the center of T , and let y be the neighbor of x in T .

If T - y has three components, two of these, say x and z, are single vertices. Then T ’ == T - { x , y , z} is a tree with at least two vertices. By the induction hypothesis, w ‘ ( V ( T ’ ) ) L 0 and, by Lemma 6 . 3 , w ’ ( { x , y , z } ) L 0. Thus w’(V(T)) 2 0.

If T - y has two components, then T ’ = T - {x,y} is a tree with at least two vertices. By the induction hypothesis, w’(V(T’)) 0 and, by Lemma 6.2, w‘({x,y}) 2 0. Thus w’(V(T)) 2 0. I

For certain classes of graphs, Lieberherr 1131 has shown that if b(G) 2 c for all graphs in such a class X, then there is a polynomial algorithm to find a bipartite subgraph B of any given G E X, with E(B) 2 c e ( G ) . Unfortunately, triangle-free graphs with maximum degree three do not constitute such a class. However, our preceding discussion can be summarized as a polynomial algorithm. We begin with any graph G E %, and any bipartite subgraph B of G.

Step 1. fails this test, replace B by BA{x} and return to step 1.

Check that &(x) L id&) for each vertex x of G. If some vertex x

Step 2. w,wI in this graph. Find vertices w,, w,, and wk of DZ3 on C, for i < j < k , with no other vertices of 0 2 3 separating them on C . Replace B by BA{w, , wi+ I , . . . , wk} and return to step 1.

Determine G[D23 U D2J. If possible, find a cycle C = w I w 2 w 3 * .

Step 3 . Determine C[D3,], its components, and all D-paths. If some D-path P = wowIw2 * * * w, has three or more vertices in 0 2 3 , let w,, w,, and wk, i < j < k , be the first three such vertices. Replace B by BA{w,, w,. I , . . . , wL} and return to step 1.

Step 4 . For each vertex x E D33, check that the three D-paths leaving x satisfy Lemma 5.2. If not, switch at most six vertices (as specified in the proof of the lemma) and return to step 1. Else calculate the revised weight w ’(x).

Step 5 . Partition each component of G [ D 3 3 ] as in the proof of Theorem 6.5. For each of these smaller sets of vertices, calculate the total of the revised weights. If the total for some set S is negative, apply the appropriate Lemma 6.2 through 6.4 depending on the size of S, then return to step 1 .

Step 6 . Display the subgraph B .

Each time the algorithm returns to step 1, E(B) has increased. Thus step 1 is executed at most e ( G ) times, and hence each step is executed at most E(G ) times.

Step 1 takes at most v (G) operations to test each vertex and, when it finds a vertex x, it takes &(x) I 3 operations to modify B . Thus the computation required for step 1 is at most O(v(G)) . (Actually, the algorithm will find a subgraph B with @) > $ E ( G ) before beginning step 2.)

Step 2 takes v ( G ) + E(G) operations to determine the subgraph and no more than E(G) operations to determine a cycle, if any. Switching takes fewer than E(G) operations.

Step 3 takes v(G) + E(G) operations to determine G[D33j, O(v(G) + E(G)) operations to find the components, and v ( G ) + E(G) operations to determine the D-paths. While finding D-paths, it is easy to test for the number vertices of DZ3 on each. Switching, if necessary, takes fewer than E(G) operations.

In step 4 all D-paths are of bounded length, thus it takes O(&) operations to check that the lemma is satisfied and to calculate w‘. Switching, if necessary, takes fewer than e (G) operations.

The partitioning of each component H , in step 5, takes O(v(H) + E(H)) = O(v(H)) operations. Partitioning of all components takes O(d3,) operations, as does checking the total revised weights. If a set S is found, let S‘ be the smallest subset of S (with CIS’] connected) which has w’(S’) < 0. Comparing S’ and its incident D-paths takes no more than some constant number of steps, depending on IS’I.

Thus the algorithm finds a bipartite subgraph B with E(B) 2 $ E ( G ) in

operations.

7. S-REGUIAR TRIANGLE-FREE GRAPHS WITH b(G) = 2 Let G be a connected, loopless, 3-regular, triangle-free graph with b(G) = :. This holds if and only if w ’ ( H , ) = 0 for all components H i i n the proof of Theorem 6.5. We shall establish some general structural properties of G and determine all possible components H i . Finally, these results will be com- bined to determine that the dodecahedron and the Petersen graph are the only such graphs.

Lemma 7.1. three and b(G) = !. Then G is 2-connected and has no multiple edges.

Let G be a connected triangle-free graph with maximum degree

Proof. That G is 2-connected follows from Lemma 3.1. Supposc u and u are vertices of C which are joined by a multiple edge. Since G is 2-connected and b(G) = ;, &(u) = 3. Let e be one of the edges joining u and u. Let B ' be a largest bipartite subgraph of G - e . If u and u are in different color classes of B ', then B ' + e is a bipartite subgraph of G. If u and u are in the same color class of B ', then (B 'A{u}) U { e } is a bipartite subgraph of G. In each case, G has a bipartite subgraph with at least E(B ') + 1 edges. Thus

contradicting the assumption that b(G) = t . We shall now consider the possible situations which may arise in Lemmas 6.1

through 6.4 if b(G) = ! for a triangle-free 3-regular graph G. For the sets of vertices considered in the proof of Theorem 6.5, each inequality in the state- ments of these lemmas must, in fact, be an equality. Figure 7.1 displays the possible neighborhoods of a vertex x of degree three in B , as determined by Lemma 5.2. (Cases (i), (ii), and (iv) of Lemma 5.2 do not occur because G is 3-regular.)

The partition of D33 given in the proof of Theorem 6.5 is unique if G is 3-regular. Suppose some component HI of G[&] has at least four vertices. Furthermore, suppose that if HI has exactly four vertices then HI has a cycle. The number of ends of D-paths in H, that are not ends of strong edges is 3v(Hl) - 2e(H1). Since G is 3-regular, each such path contributes at least -2 to the revised weight of its end(s) in HI. Therefore,

w'(V(H,)) 1. 3 v ( H , ) - 2(3v(Hl) - 2&(H,))

= -3v(H,) + 4&(H1)

1 -3v(H,) + 4(v(H,) - 1)

= v(H,) - 4

(equality only if H, is acyclic)

(equality only if v(H,) = 4) .

Thus w'(&) > 0 and b(G) > $, which contradicts the choice of G. Since the partition is unique and the components Hi all have at most four ver-

tices, we can enumerate all possible components Hi of G[D33]. Later, we shall consider how these components may be attached to each other. Let H be any component of G[D,,]. We consider four cases, depending on the number of vertices in H.

w"x)=- I

case (iii)

case (v)

case (vi 1 FIG. 7.1

Case (i): v(H) = 1. There is only one possible component with exactly one vertex (see Figure 7.1, Case (vi), first diagram). In the case in which each component of G[D33] has exactly one vertex, Malle 1171 has shown that G is either the Petersen graph or the dodecahedron. Thus we need consider only those cases in which at least one component has two or more vertices. However, for the sake of completeness, we shall deal with this case as well as those in which v(H,) 5 2, for some i.

Case (ii): v(H) = 2. Suppose a component H has exactly two vcrtices, say .r and y. Then either w ' ( x ) = w'(y) = 0 or, without loss of generality, w'(x ) =

- 1 and w ’( y) = + 1. There are six combinations of adjacent vertices x and y with w’(x) = w’(y) = 0. Figure 7.2 depicts these six cases. Some are forbidden because there is a set X (indicated by a closed curve) for which &@AX) > &(B). Thus, only those configurations shown in the last three diagrams are actually possible as components of G[D,,] and their incident D-paths.

There are two combinations of adjacent vertices x and y with w ‘ ( x ) = - 1 and w‘ (y ) = 1. One of these two is forbidden (see Figure 7.3). In the other,

Q I’ NS I

FIG. 7.2 b

FIG. 7.3

the vertices x and y are not the only vertices in the component H , but we include it here as a building block for components with three or four vertices.

Case (iii): v(H) = 3 . Suppose a component H has exactly three vertices, x, y , and z. Since w’({x,y,z}) = 0, w’({x,y}) 2 0, w‘({y,z}) 2 0, and ~ ‘ ( x ) , w ’(.y), w ’(z) are all at least - 1 , the only possibilities for (w ’(x), w ’ ( y), w ‘(z)) are (- 1 , 1, 0) , ( - 1,2, - I ) , ( O , O , 0), and (0, 1, - 1). There is no way to join with weights zero. Thus the only cases to considcr are, without loss of generality, ( - 1. 1,O) and (- I , 2, - I ) . In the first case, the vertices x and Y must be as depicted in Figure 7.3 (b). The first diagram of Figure 7.4 shows that at least two of the edges of E ( G ) - E ( B ) must join vertices within the diagram. (We can switch on the set X indicated or on X U { N } . ) There are only two choices for these edges, and these are shown in Figures 7.4 (b) and (c). The vertices labeled xI, x2, y I , z I , and z2 in F , and F2 are not necessarily distinct.

The sequence (- 1,2, - 1 ) leads to only one possibility. This is displayed in Figure 7.5.

Case (iv): u ( H ) = 4 . Suppose H has exactly four vertices, x, y, z, and a. If N(a) = { x , y , z}, then there is only one configuration (see Figure 7.6). Thc set X of vertices indicated in the diagram forces x ’ and y” to be joined by an edge of E ( G ) - E(B) . By symmetry, there must also exist edges y’z‘’ and z’x’‘. Thus the situation is as shown in the second diagram.

If xyzu is an induced path in G then the restrictions on the sums of subse- quences of ( ~ ‘ ( x ) , rv’(y), w‘(z), w ’ ( a ) ) imply the only possible vectors are

FIG. 7.5. F3.

Y2 FIG. 7.6

(O,O,O,O), (O,O, 1 , - l ) , ( - 1 , 1,0,0), and ( - 1 , 1 , 1 , -1). Of these, only ( - 1, 1, I , - 1) is constructible. The edges marked e in the first diagram of Figure 7.7 must be the same edge, as must the edges marked e ' . Thus there is only one configuration for this sequence.

If G [ { x , y , z , a } ] is not a path or a K 1 , 3 then it contains a cycle, and w ' ( V ( H ) ) > 0. Thus we have described all components H for which v (H) = 4.

Before combining the configurations allowed for H and its incident D- paths, we observe the following closure properties common to all of the above configurations.

(cl): Let uab be a path in B with u E Dj3 and a , b E D23, and let e be the edge of E ( G ) - E ( B ) with end b. Then there is a 5-cycle of G containing both v and e .

FIG. 7.7

(c2): Let R be a path of length two or three in D33. Then R is contained in a 5-cycle of G.

We shall assume that we have chosen B from among all bipartite subgraphs of G which contain $ of the edges of G, so that the largest component of G[D3)] has as marly vertices as possible. Let H be such a component of G[D,,]. We have already observed that H has at most four vertices and classified the possibilities for H and its incident D-paths. We shall now prove that G is the Petersen graph or the dodecahedron. Again, we consider four cases, depending on the number of vertices of H.

Case (i): v(H) = 1. Every component of G[D3J corisists of a single vertex. Let x and y be vertices of Da at distance two in B . Figure 7.8 (a) displays the various possibilities, with x and y connected by one, two or three D-paths in B . For each of these, a set X has been indicated for which E ( B A X ) > E(B) . Since G has no triangle, x and y cannot be connected by more than one D-path and, if they are connected by exactly one D-path, then some edge of E(G) - E(B) has both ends in X . This situation is depicted in Figure 7.8 (b). The vertices which are labeled r , s, t , and u in this diagram are not necessarily distinct, although r # u, as we have just seen, and r # s, since G has no triangle. If r # t , then, with X as indicated, & @ A X ) = E(B) and BAX has a component of G[D,,] with

(d 1 FIG. 7.8

at least two vertices, r and y’. This contradicts the choices of B and H . Thus r = t and, similarly, s = u . The revised configuration is shown in Figure 7.8 (c).

Since G has no triangle, edges u and b are distinct, as are edges b and c. If a and c are also distinct then, for the set X indicated in Figure 7.8 (c), & @ A X ) > E ( B ) , contradicting the choice of B . Thus a = c.

The revised configuration is displayed in Figure 7.8 (d). If the edgesfand d, indicated in this diagram, are distinct, then & @ A X ) = E ( R ) . But BAX has a component of G[D3, ] with at least two vertices, specifically s and x ’ . This contradicts the choices of B and H . Thus b = d. Similarly b = e, and hence, r r = s ’ . Every vertex of this configuration now has degree three within the configuration. Therefore, we have described G cornplctely, and G is the Petersen graph.

Case (ii): v(H) = 2. The three possibilities for H are depicted in Figure 7.2. In each of these diagrams, vertices a , b, q , r , and s have been labeled. Let B ’ = B A { a , b} for each of these configurations. Then E(B I ) = e(R). Further- more, the only vertices of B ‘ which have smaller degree in B ’ than in B are y and s. But s @ {q ,x} , since G has no triangle, y @ {q ,x} , since y is in a different color class than q and x, and r e { y , s}, since d B ( r ) = 2 and dB( y) = &(s) = 3 . Thus {q, r , x } fl { s , y } = 0, and q, r , and x all have degree three in B ‘ , contradicting the choices of B and H .

Case (iii): v(H) = 3 . There are three possibilities F , , F?, and F , for H , as depicted in Figures 7.4 and 7 . 5 . In Figure 7.4, vertices u and b have been labeled in each of F I and F 2 . We consider first F I . Let E l = B A { u , b } . Then & ( E l ) = &(B). We shall show that y , z , c, and z I each has degree three in B , , contradicting the choices of B and H . The only vertices of E l which have smaller degree in B , than in B are x and x?. We need only show that z I { x , x z } . But z I and x2 are in different color classes of B and, if z I = x , then xac is a triangle in G. Thus each of y, z , c , and zI has degree three in E l .

Suppose, now, that H and its neighboring D-paths are represented by F z , and let B2 = B A { a , b } . Then 4 B 2 ) = E ( B ) , and z and z2 are the only vertices of degree two in B2 that are of degree three in B . Furthermore, { z , z z } f l { c , d , x , y } = 0 and, thus, c, d , x, and y are all of dcgree three in B 2 , contradicting the choices of B and H .

The last possibility for H and its D-paths is F3 of Figure 7.5. Let B3 = BA{a, b , x , c}. Then E(&) = &(B) , and b, e, j , and z all have degree three in B 3 , contradicting the choices of B and H .

Thus, the largest component of D33 cannot have exactly three vertices.

Case (iv): v(H) = 4: The two possible configurations are depicted in Figures 7.6 and 7.7. We shall consider Figure 7.6 first. By the closure property (cl), x2 = zI, y2 = XI, and z2 = yI. Figure 7.9 displays the configuration with these vertices identified. The neighbors x3, y3 , and z3 of x,, y I , and z l , respec-

FIG. 7.9

tively (as labeled in the diagram), each has degree three in B . By the closure property (c2), xI and x3 must be contained in a 5-cycle. Thus, without loss of generality, x3 = y3. Similarly z3 = x3, and G is the dodecahedron.

The second possible configuration is illustrated in Figure 7.7. By the closure property (cl) , a2 = y , , x? = z , . and there are edges a lz l 2nd x ly l in B . The configuration, with these new relationships is shown in Figure 7.10. The third neighbors x3 and a3 of x , and a,, respectively, have degree three in B. Thus, by the closure property (c2), x2, a, and u3 are contained in a common 5-cycle. This is only possible if u3 = x, and G is the dodecahedron.

In all possible cases we have shown that G must be either the Petersen graph or the dodecahedron. Thus we have the following theorem.

Theorem 7.2. exactly t . Then G is either the Petersen graph or the dodecahedron.

Let G be a triangle-free 3-regular graph with bipartite density I

FIG. 7.10

For the more general case with vertices of degree two and three, we make the following conjecture.

Conjecture 7.3. Let G be a triangle-free graph with maximum degree at most three and bipartite density exactly 2. Then G is one of the seven graphs displayed in Figure 7.11.

Since the Petersen graph and the dodecahedron are the only 3-regular graphs with bipartite density exactly !, it might be hoped that Theorem 6.5 could be improved for “large” graphs. The following two examples set limits on any such improvement.

Example 7.5. Let H be a cycle and let G be the graph constructed from H by replacing each vertex of H by the graph F depicted in Figure 7.12. (Note that we do not consider the edges e and f in Figure 7.12 as part of F . ) Then G is 2-connected, 3-regular, triangle-free, and b(G ) 5 2.

Proof. Each copy of F has a 2-covering by 5-cycles. Thus, by Corol- lary 2.5, b(F) 5 2. Also, we note that E ( F ) = 20, E(H) = v ( H ) , and E(G) = 21v(H). Hence, by Lemma 2.4,

FIG. 7.11

FIG. 7.12

Example 7.6. Let H be a 3-connected 3-regular graph and let G be the graph constructed from H by replacing each vertex of H by the graph F displayed in Figure 7.13. Then. G is 3-connected, 3-regular, triangle-free, and h(G) 5 ?i.

Proof. Each copy of F has a 2-covering by 5-cycles. Therefore, by Corol- lary 2.5, b(F) 5 4 . Also,

E ( F ) = 15, &(If) = ; v ( H ) ,

FIG. 7.13

&(G) = $ v ( G ) = ? v ( H ) .

Thus, by Lemma 2.4,

Hence, even the requirement that G be 3-connected would not significantly raise the lower bound on b(G) given in Theorem 6.5.

8. m-COVERINGS BY 5-CYCLES

If we restrict our attention to triangle-free graphs with maximum degree three which have an m-covering by 5-cycles, then, for any such graph,

b(G) = $.

Thus, by enumerating the triangle-free graphs with maximum degree three and bipartite density exactly :, we would be enumerating those triangle-free graphs G with maximum degree three which have an m-covering by 5-cycles. The following lemma shows that only triangle-free graphs can admit such a covering.

Lemma 8.1. triangle, then G has no m-covering by 5-cycles.

Let G be a graph with maximum degree three. If G contains a

Proof. Let T = {el , e2, e3} be the set of edges of a triangle in G, and let T ' be the set of edges of G - T which are incident with an edge of T . Suppose X is an m-covering of G by 5-cycles. Let Xr be the set (with multiplicities, if necessary) of 5-cycles in X which contain an edge of T . Any 5-cycle in X r contains at most two edges of T and exactly two edges of T ' . Thus, IT'I 2 (TI, and each vertex incident with an edge of T has degree three in G. Figure 8 .1 displays the portion of the graph containing T and T ' with T ' = {fl,f2,&} and N ( T ) - V ( T ) = { u l , u 2 , u 3 } , as labeled. Each of the edges fI,f2,f3 is covered exactly m times by cycles of X r . Thus 21Xr1 = 3m. Each of the edges e l , ez, e3 is also covered m times by cycles of X T . If there would be a cycle of X r containing exactly one of the edges e l , e2 , e3 , then 21XTl > 3m. Thus each cycle of X T contains exactly two of the edges e l , e2, e3. Let C be a 5-cycle in X T containing e2 and e3 . Then {f2, e3, e2,&} E C, and C has exactly

"I FIG. 8.1

FIG. 8.2

one more edge, namely u2u3. Thus uzuB and, similarly, u luz and uIu3 are edges of G. Therefore G is the triangular prism depicted in Figure 8.2. But this graph has no m-covering by 5-cycles, since X - X T would have to cover the triangle G - T - T' exactly m/2 times.

In Section 7, we proved that the only 3-regular triangle-free graphs with b(G) = $ are the Petersen graph and the dodecahedron. This result leads to the following theorem.

Theorem 8.2. Let G be a 3-regular graph which has an m-covering by 5- cycles, for some positive integer m. Then G is either the Petersen graph or the dodecahedron.

Proof. Immediate from Corollary 2.5 and Lemma 8.1.

Since a 3-regular graph which contains a triangle cannot have an m-covering by 5-cycles, we may restate Conjecture 7.3 as follows.

Conjecture 8.3. Let G be a graph with maximum degree at most three. Suppose that G has an m-covering by 5-cycles, for some positive integer m. Then G is one of the seven graphs depicted in Figure 7.11.

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Largest bipartite subgraphs in triangle-free graphs with maximum degree three

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