The number of shortest cycles and the chromatic uniqueness of a graph

The Number of Shortest Cycles and the Chromatic Uniqueness of a Graph

C.P. Teo and K.M. Koh DEPARTMENT OF MATHEMATICS

NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE

ABSTRACT

For a graph G, let g(G) and aJG) denote, respectively, the girth of G and the number of cycles of length g(G) in G. In this paper, we first obtain an upper bound for a,(G) and determine the structure of a 2-connected graph G when a,(G) attains the bound. These extremal graphs are then more-or- less classified, but one case leads to an unsolved problem. The structural results are finally applied to show that certain families of graphs are chro- matically unique.

1. INTRODUCTION

Let P(G, A), or simply P(G), denote the chromatic polynomial of a (simple) graph G (see [6]). Two graphs G and H are said to be X-equivalent, written G - H, if P(G, A) = P(H, A). A graph G is said to be X-unique (see [l]) if for any graph H such that H - G, we have H = G.

For a graph G containing a cycle, the girth g(G) of G is the length of a shortest cycle in G. Let a,(G) denote the number of cycles of length g (= g(G)) in G. It then follows from Whitney’s broken-cycle theorem [ll] that if G and H are graphs containing cycles such that G - H, then g(G) = g ( H ) and also aJG) = a J H ) (see also [5]).

What can be said about the quantity a,(G)? In Section 2, we first establish the following upper bound for r,(G). If G is a 2-connected graph of order n, size m, and girth g , then

-(m - n + l),

-(m - n + 1)

when g is even ;

wheng is odd.

CCC 0364-9024/92/010007-09$04.00

I: U,(G) 5

Journal of Graph Theory, Vol. 16, No. 1. 7-15 (1992) 0 1992 John Wiley & Sons, Inc.

8 JOURNAL OF GRAPH THEORY

We then proceed to show that for even g, the equality holds iff G satisfies the condition

(*) every two edges of G are contained in a common shortest cycle C,.

And for odd g, the equality holds iff G satisfies the condition

(#) every vertex and every edge of G are contained in a common short-

In Section 3, we attempt to classify all graphs satisfying these conditions. The problem of determining whether a graph is X-unique is, in general,

very difficult, and no general method for finding such a graph is known. It is noted that the above structural results can be applied to show that a number of families of graphs are X-unique. We give two simple examples in the final section as an illustration of a way in which these structural results are used.

est cycle C,.

2. THE NUMBER OF SHORTEST CYCLES AND STRUCTURAL THEOREMS

In this section, we give the uper bound for a,(G) and determine the structure of G when aJG) attains the bound.

We first consider the case when g(G) is even. Let G be a graph with g(G) = 2f, f 2 2, and let CZJ: u l u z . . . U2JU1 be a cycle of length 2f in G. Two edges u,u,+l and U , U , + ~ (mod 2f) of CZf are called opposite edges of CZf if d(u,, u,) = f i n c2J. Two edges el and e2 of G are called opposite edges of G if they are opposite edges of some CZJ in G. In this case, e l is called an opposite edge of e2, and vice versa.

Lemma 1. opposite edges of G, there is exactly one C , containing them.

Let G be a graph with g(G) = 2f, f 2 2. Then, for every two

ProoJ: Let e 1 , e 2 be two opposite edges of G. Then e l and e2 are opposite edges of some CZf. If e l and e2 are contained in another c2J, then it is easy to see that the union of these two cycles contains a cycle of length at most 2f - 1, a contradiction. I

Lemma 2. Suppose that G is a connected graph with g(G) 2 2f, f 2 2. Let u E V(G) and let

W = {U E V(G) I ~ ( u , u ) I f - l}.

Then the subgraph ( W ) of G induced by W is a tree.

Prooj Since G contains no cycle of length less than 2f, each vertex u of W is connected to u by a unique path P, of length d(u,u) . If u f u,

SHORTEST CYCLES AND CHROMATIC UNIQUENESS 9

let e,, be the edge of P, incident with u. Clearly the subgraph induced by {eu I u f K} is a spanning tree T of ( W ) such that dT(u, u) = dc(u, u) for each u. If u I u 2 is any other edge of ( W ) , then G contains a cycle of length at most 1 + d ( u , u l ) + d(u ,u , ) I 2f - 1, a contradiction. I

Lemma 3. graph of G. Write

Let G be a connected graph with 6(G) 2 2 and H be a sub-

Then

(9 q 2 P , (ii) q = p iff H = G.

Proof: If H f G, then the multigraph M obtained from G by contract- ing all edges in H is a connected multigraph with at leastp + 1 vertices and exactly q edges, and so q 2 p + 1 unless M is a tree. But an end vertex in M must be in H, and so if M is a tree, then IV(M)I 2 p + 2 and again q r p + l . I

We are now in a position to prove our first structural result.

Theorem 1. withg(G) = 2f, f 2 2. Then

Let G be a 2-connected graph of order n and size m, and

(i) every edge of G is contained in at most m - n + 1 cycles C Z ~ , (ii) ay(G) I (m(m - n + 1))/2f, and the equality holds iff G satisfies

condition (*).

Proof: For each e E E(G), let a(e) denote the number of C2f’s containing e . Fix an arbitrary edge e = ab in G. Let H denote the subgraph of G induced by the edges of the a(e) cycles Czf containing e. (Thus, for each e’ E E ( H ) , e’ + e, { e , e ’ } is contained in a common Czf.) Let Z(P) denote the length of a path P, and let

T, = {u E V ( H )

Tb = {U E I/(H)

there exists in H an a - u path P with Z(P) I f - 1 and b @ V(P)} ,

there exists in H a b - u path Q

with I ( Q ) I f - 1 and a @ V(Q)},

, % a,u) = f - l}, and A = {u E T, Id,

B = {u E Tb I d ~ ( b , U ) = f - I } .

Since g(G) = 2f, T, n Tb = 0 (in particular, A n B = 0). Let

X = {x E E ( H ) I x joins a vertex in A to a vertex in B}.


Clearly, X is the set of opposite edges of e , and by Lemma 1, 1x1 = a(e). Letp = lV(G)\V(H)l and q = IE(G)\E(H)I. Then IT, U Tbl = IV(H)I =

n - p, and the union of the subgraphs (T,) and (Tb), being a forest with two components by Lemma 2 , contains exactly n - p - 2 edges. Thus, by counting the size of G, we have

rn = IE(G)\E(H)I + IE(H)I = q + 1 + (n - p - 2) + 1x1, and so

a(e) = 1x1 = m - n + 1 - (q - p).

By Lemma 3, we have a(e) I rn - n + 1, which proves (i), and a(e) = rn - n + 1 e H = G e {e,e'} is contained in a C2f for each e' E E(G), e' f e.

Now azf(G) = (1/2f)E(a(e)le E E(G)) 5 (rn/2f)(rn - n + l), and the equality holds iff every two edges of G are contained in a common Czf. This proves (ii). I

We now consider the case wheng(G) is odd. Let G be a graph withg(G) = 2f + l,f I 1, and let CY+, : u l u z . . . u2f+lul be a cycle of length 2f + 1 in G. An edge U , U , + ~ (mod 2f + 1) is called the opposite edge of a vertex u, in CZf+ , if d(u,, u , ) = f = d(u,, u , + ~ ) in Czf+l. An edge e of G is called an opposite edge of a vertex u if it is the opposite edge of u in some Crf+l. By an argument similar to the proof of Lemma 1, the following observation can be proved.

Lemma 4. Let G be a graph with g(G) = 2f + 1, f 1 1. Then, for each vertex and each of its opposite edges, there is exactly one C2f+l containing them.

Theorem 2. Let G be a 2-connected graph of order n and size rn, and withg(G) = 2f + 1. Then

and the equality holds iff G satisfies condition (#).

Proo$ For each u E V(G), let a(u) denote the number of C2f+l's containing u. Fix an arbitrary vertex u in G, and let H denote the subgraph of G induced by the edges of the a(u) cycles C2f+l containing u. (Thus for each e E E(H) , {u, e} is contained in a common C2f+l .) Let X ( C E(H) ) be the set of opposite edges of u. Then 1x1 = a(u). Let H ' = (E(H)\X). We observe that H ' is a tree, since if T is the spanning tree of H ' formed as in


Lemma 2, and H ' has an edge e not in T, then H ' contains a cycle of length at most 2f or a cycle of length 2f + 1 in which the opposite edge of u is e, which is not in X, and neither of these is possible.

Let p = lV(G)\V(H)J and q = IE(G)\E(H)I. By counting the size of G, we have

and so a(u) = (m - n + 1) - (q - p ) I m - n + 1 by Lemma 3. More- over, the equality holds * q = p tj H = G * {u,e} is contained in a common CZf+] , for each e E E(G). Hence,

and the equality holds iff every vertex and every edge of G are contained in a common C 2 f + ] . I

3. CLASSIFICATION OF EXTREMAL GRAPHS

It is clear from the proof of Theorem 1 that if G has girth g = 2f, then G satisfies condition (*) iff every vertex of G is within distance f - 1 of a given edge e. Similarly, it is clear from the proof of Theorem 2 that if G has girth g = 2f + 1, then G satisfies condition (#) iff every vertex of G is within distance f of a given vertex u. In each case, an r-regular graph with this property is called an (r, g)-Moore graph if r L 3 and g L 5. Thus it is easy to see that the following result holds.

Lemma 5. An r-regular graph G of girth g satisfies condition (*) (if g is even) or (#) (if g is odd) iff one of the following holds:

(i) r = 2, g 2 3, and G is a g-cycle; (ii) r L 3, g = 3, and G is the complete graph K,;

(iii) r L 3, g = 4, and G is the complete bipartite graph Kr,r; (iv) r 2 3, g 2 5, and G is an (r,g)-Moore graph.

We propose to use the term (r, s, g)-semi-Moore graph to denote a bipartite graph of even girth g L 6 with partite sets X and Y such that every vertex of X has degree r , every vertex of Y has degree s, and r > s 2 3. An interesting unsolved problem, analogous to that of classifying the Moore graphs, is to classify the semi-Moore graphs. We have not even been able to establish whether any semi-Moore graphs exist.


A nonregular graph G is said to be semiregular if there exist two distinct integers r and s such that deg u E {r,s} for each u in V(G). A k-uniform subdivision of a graph G is one obtained from G by replacing each edge of G by a path of length k. A uniform subdivision of G is a k-uniform subdivision of G for some k 2 1.

Very recently, Honiobono and Peyrat investigated in [3] the structures of graphs satisfying (*) with motivation different from ours. They showed that if G is a graph with g(G) = 2f satisfying (*), then

(1) G is bipartite; (2) deg u = deg u for all u, u in V(G) such that d(u, u) = f ; and (3) If deg u 2 3, then deg uI = deg u2 for all u 1 , u 2 in V(G) such that

Using (2) and (3), they further showed that G is a uniform subdivision of a regular graph of even girth and degree at least 3, a semiregular graph of even girth and minimum degree at least 3, or a multigraph of order 2. This leads to the following theorem.

d ( U , U l ) = d ( U , U 2 ) .

Theorem 3. (*) iff G is a uniform subdivision of one of the following:

Let G be a graph of even girth. Then G satisfies condition

(i) an (r , g)-Moore graph where r 2 3 and g = 6, 8, or 12; (ii) an (r, s, g)-semi-Moore graph where r > s 2 3, g is even, and g 2 6; (iii) a complete bipartite graph Kr,s where r 2 s 2 3; and (iv) a multigraph of order 2.

Prooj Note that if G and H are graphs such that G is a uniform subdivision of H, then G satisfies (*) iff H satisfies (*). Thus it suffices to prove that the graphs listed in (i)-(iii) are precisely the same as the regular and semiregular graphs of even girth satisfying (*) and having no vertices of degree 2. This follows from Lemma 5, a known result about the existence of Moore graphs (see Theorem 3 in [12]), and the observation that a graph of girth 4 satisfies (*) iff it is a complete bipartite graph. I

We now consider graphs of odd girth g satisfying condition (#). We first show that such graphs are always regular. The proof given below turns out to be similar to that given in [S].

Lemma 6. tion (#). If u, u E V(G) such that d(u, u) = f , then deg u = deg u.

Let G be a graph withg(G) = 2f + 1,f 2 1, satisfying condi-

ProoJ: Let u , u E V(G) such that d(u,u) = f . We first observe by assumption that given any two u - u paths Q, , Qz with /(el) = Z(Qz) = f + 1, QI f Q 2 iff QI and QZ are internally disjoint. Let P be a u - u path of lengthf. By assumption, such P is unique. We now claim that each edge e incident with u, e $? E(P), is contained in a unique u - u path of


lengthf + 1. Indeed, by hypothesis, there is a cycle C2f+l = P’ U Q’ containing e and u, where P‘ and Q’ are two internally disjoint u - u paths with e E E(P‘) . Since d(u,u) = f , either Z(P’) = f or &Q’) = f. By the uniqueness of P, Z(P’) f f and so Z(P’) = f + 1. This shows that e is contained in a u - u path P‘ of lengthf + 1, and by the above observation, such P‘ is unique. It now follows from the claim that deg u I deg u. By symmetry, we have deg u = deg u. I

Corollary. tion (#), then G is regular.

If G is a graph with g(G) = 2f + 1, f 1 1, satisfying condi-

ProoJ To show that G is regular, take any two adjacent vertices u, u of G. Then uu is contained in a C2f+l. Let w E V(C,+,) such that uu is the opposite edge of w in C2f+l. Then d(u, w) = d(u, w) = f and thus deg u = deg w = deg u by Lemma 6. I

Let G be a graph of odd girth g satisfying (#). Then G is regular by the above corollary. By Lemma 5 and a known result about the existence of a Moore graph (see [12]), we have the following result.

Theorem 4. Let G be a graph of odd girth. Then G satisfies condition (#) iff G is a complete graph, an odd cycle or an (r,5)-Moore graph where r = 3, 7, or (possibly) 57.

4. APPLICATIONS

In this final section, we shall show how the results obtained in the preceding sections can be used to prove the chromatic uniqueness of certain graphs.

By employing the notion of simplified adjacency matrix of a bipartite graph, Teo and Koh proved in [9] the conjecture of Salzberg et al. [7] that every complete bipartite graph K p , q , q 2 p L 2, is X-unique. We now offer a new proof of this result by applying Theorem 1.

Lemma 7. Let G and H be two X-equivalent graphs. Then

Remarks. Results (iii) and (iv) follow from Whitney’s broken-cycle theorem (see [ll] and [5]) . Results (v) for k = 2 was proved independently in [13] and [lo].


Theorem 5. Let G be a 2-connected graph of order n = p + q and size rn = p q such that g(G) = 4 and a4(G) = (p2)(;) where q 2 p 2 2. Then G = Kp,q .

Prooj Observe that

rn 1 -(rn - n + 1) = -pq(pq - p - q + 1) = g 4

Thus by Theorem l(ii), G satisfies condition (*). By a result of Homobono and Peyrat (see Section 3), G is bipartite. Condition (*) now requires G must be a complete bipartite graph. Let X,Y be the two partite sets of G with 1x1 = p + r and IYI = q - r for some r 2 0. Then ( p + r)(q - r) = 1x1 IYI = p q and so r(q - p - r) = 0, which implies that G z Kp,q . I

By Theorem 5 and Lemma 7, we have

Corollary. The complete bipartite graph Kp,q is X-unique for all q 2 p 2 2.

By applying his newly developed algorithm [4], Read verified that the Petersen graph (the unique (3,5)-Moore graph) is X-unique. A mathemati- cal proof of this result has also been given in Chia [2]. The Hoffman- Singleton graph (see [12]) is the unique (7,5)-cage. It is easy to check that it is the (7,5)-Moore graph.

Theorem 6. (i) Let G be a 2-connected graph of order n = 10, size rn =

15 and girth g = 5 , and with as(G) = 12. Then G is the Petersen graph. (ii) Let G be a 2-connected graph of order n = 50, size rn = 75 and

girth g = 5, and with as(G) = 1260. Then G is the Hoffman-Singleton graph.

Proo$ (i) Observe that (n/g)(rn - n + 1) = 12 = as(G). By Theorem 2, G satisfies condition (#). By Theorem 4, G is a (3,5)-Moore graph. Thus G is the Petersen graph.

The proof of (ii) is similar. I

Corollary. The Petersen graph and the Hoffman-Singleton graph are X-unqiue.

ACKNOWLEDGMENT

The authors would like to thank one of the referees for giving neater proofs to some of the results in this paper, and his critical comments and helpful suggestions, which led to this improved version.


References

[l] C.Y. Chao and E.G. Whitehead, Jr., On chormatic equivalence of graphs. Lecture Notes Math. 642 (1978) 121-131.

[2] G.L. Chia, The Petersen graph is uniquely determined by its chromatic polynomial. Preprint (1988).

[3] N. Homobono and C. Peyrat, Graphs such that every two edges are contained in a shortest cycle. Discrete Math. 76 (1989) 37-44.

[4] R. C. Read, An improved method for computing the chromatic polynomials of sparse graphs. Research Report CORR 87-20, Department of Combinatorial and Optimization, University of Waterloo (1987).

[5] R.C. Read, Recent advances in chromatic polynomial theory (to appear).

[6] R. C. Read and W.T. Tutte, Chromatic polynomials. Selected Topics in Graph Theory 3, Academic Press, New York (1988), 15-42.

[7] P.M. Salzberg, M.A. Lbpez, and R.E. Giudici, On the chromatic uniqueness of bipartite graphs. Discrete Math. 58 (1986) 285-294.

[8] R.R. Singleton, There is no irregular Moore graph. Am. Math. Monthly 75 (1968) 42-43.

[9] C. P. Teo and K. M. Koh, The chromaticity of complete bipartite graphs with at most one edge deleted. J. Graph Theory 14 (1990) 89-99.

[lo] E .G. Whitehead, Jr., and L.C. Zhao, Cutpoints and the chromatic polynomial. J. Graph Theory 8 (1984) 371-377.

[ll] H. Whitney, The colouring of graphs. Ann. Math. 33 (1932) 688-718. [12] P. K. Wong, Cages-A survey. J. Graph Theory 6 (1982) 1-22. [13] D. R. Woodall, Zeros of chromatic polynomials. Combinatorial Sur-

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The number of shortest cycles and the chromatic uniqueness of a graph