Subdivisions of Graphs with Large Minimum

~

Carsten Thomassen Degree

THE TECHNlCAL UNlVERSlTY OF DENMARK LYNGBY. DENMARK

ABSTRACT

We prove a theorem on path systems implying the conjecture of BollobBs that there exists a function f ( k , rn) (where k and rn are natural numbers) satisfying the following: For each graph G of minimum degree at least f (k , rn) there exists a graph H of minimum degree at least k such that G contains the graph obtained from H by subdividing each edge rn times.

1. INTRODUCTION

A fundamental and useful result of Mader (31 asserts that each graph of sufficiently large minimum degree contains a subdivision of a complete graph of prescribed order. The author [4] proved that the subdivision can even be chosen such that each edge of the complete graph is subdivided a prescribed odd number of times modulo k (where k is a fixed natural number), extending thereby also the result of Bollobis [ I ] on cycles modulo k. The subdivision cannot be chosen such that each edge is subdivided a prescribed number of times because there exist graphs of any prescribed minimum degree and girth. Instead Bollobis [2] made the conjecture described in the Abstract and pointed out that this generalizes his result in [I]. The purpose of this note is to prove that conjecture.

The union of k paths of length rn having precisely an endvertex u in common pair by pair is called a (k,rn)-fan with center u. The set of neighbors of a vertex set A in a graph is denoted N(A) . We shall use the following variant of Hall’s theorem: If G is a bipartite graph with partite sets A and B, then G has a collection of pairwise disjoint (k, 1)- fans such that A is the set of centers if and only if, for each subset A‘ of A, we have lN(A’)I 2 kp‘(. This variant follows from the standard version of Hall’s theorem (where k = 1) by replacing each vertex x of A by k vertices each having the same set of neighbors as x .

Journal of Graph Theory. Vol. 8 (1984) 23-28 0 1984 by John Wiley & Sons, Inc. CCC 0364-9024/84/010023-0684.00

24 JOURNAL OF GRAPH THEORY

2. DOMINATION IN BIPARTITE GRAPHS

We first prove a result asserting that, if each vertex of one of the partite sets, say B, of a bipartite graph is joined to many vertices of the other partite set A, then A has a small subset A' such that almost all vertices of B are joined to many vertices of A'. More precisely, we have

Theorem 1. Let G be a bipartite graph with partite sets A and B and k a natural number such that each vertex of B has degree at least k4 + k2. Then there are subsets A' and B' of A and B, respectively, such that

and each vertex of B\B' is adjacent to at least k vertices of A'.

Proof. We label the vertices of A as follows: Having already defined a , , a2 , ..., a; (i 3 0), we define a;+, as a vertex which, among the vertices of A\{al , a2, ..., ai} , has maximum degree in G - ({a, , a2, ..., a;} U N({a , , a2, ..., ai})). Let q denote the smallest integer greater than or equal to lAl/k2. and put A , = {a , , a2, ..., aq}. Let d be the degree of aq+, in G - (A, U N(A,)) and put B, = N(A,). By the way in which the vertices of A are labeled we have

The number of edges from B\B, to A\A, is at least

and the same number of edges counted from A\AI is at most

so

Let B; be the set of those vertices of B , which are adjacent to at least k vertices of A , . Then each vertex of B,\B; is joined to at least k4 + k2 - k 5 k 4 vertices of A\AI and so we can repeat the above argument with B,\B; instead of B and A\AI instead of A. In this way we obtain sets B2 and A2 such that

SUBDIVISIONS OF GRAPHS 25

and each vertex of B2 is joined to A 2 . Proceeding in this way we obtain disjoint sets A , , A * , ..., Ak, B;, B;, ..., B;- ,, and a decreasing sequence B 2 B I 2 Bz 2 .... 2 Bk such that for each j = 2, 3, ..., k,

and such that each vertex of Bi- I is joined to at least k vertices of Aj- I ,

and each vertex of Bj is joined to Aj. Now the theorem follows with

i.e.,

3. COLLECTIONS OF FANS WITH ENDVERTICES OF LARGE DEGREE

Using Theorem 1 we prove a result on the existence of certain fans in graphs of sufficiently large minimum degree.

Theorem 2. For any natural number k and any non-negative integer m, there exists a natural number g(k, m) such that every graph of minimum degree at least g(k, m) contains a set A of vertices and a collection of klAl (k , m)-fans which are pairwise disjoint and disjoint from A such that all endvertices of these fans are joined to at least k vertices of A .

Proof(by induction on m). We have g(k, 0) s 2(k + 1)‘ + 2(k + 1)’ because every graph of minimum degree at least 2(k + + 2(k + 1)’ contains a bipartite graph with partite sets A and B , of minimum degree at least (k + 1)‘ + (k + 1)’. We choose the notation such that 1B1 2 IAI and apply Theorem 1 (with k + 1 instead of k).

Now suppose that m 2 1 and that the theorem has been proved for all smaller values of m. We consider first the case m odd, i.e., m =

29 + 1 where q is a non-negative integer. Let k be any natural number. We show that g(k. rn) s g(K , q) where K = (8k)‘ + (Sk)’. So we consider a graph G of minimum degree at least g(K , 4). By the induction hypothesis, G contains a set A of vertices and a collection of KIA1 (K, q)-fans FI, F2, ..., F+I which are pairwise disjoint and disjoint from A such that all endvertices of each Fi are adjacent to at least K vertices of A. For each i = 1, 2, ..., m(, let Si be the set of endvertices of Fi (if q = 0, we put Si = Fi). We choose A and the fans Fi such that \A\ is minimum.

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We claim that G contains a collection of (K , I)-fans F ; , F ; , ..., F ~ I whose centers are precisely the set A and whose endvertices are in S , U S2 U - - - U SwI such that no Si (1 s i Q KWI) contains more than one endvertex of F ; U F ; U 0 . . U FLl. For if this were false, then by the variant of Hall's theorem mentioned in the introduction, A would contain a subset A' such that N(A') intersects less than KIA'] of the sets S , , S2, ..., S o l . But then A\A' together with those fans Fi ( 1 Q i S

NAI) which do not intersect N(A') would contradict the minimality of A. This proves the claim that F ; , F ; , ..., FI;., exist.

For each endvertex x of F ; U F ; U -.. U Fh, we consider the fan Fi (1 s i Q KIAI) that contains x and we let P, denote a path of Fi that connects x with another vertex of Si. Then the union of F ; U F ; U ... U Fhl and the paths P, forms a collection of (K, m)-fans F;, F; , ... , FGl whose endvertices form a set B C SI U S2 U .-- U SKBI. By Theorem 1, there are subsets A' and B' of A and B, respectively,

such that

and each vertex of B\B' has at least 8k neighbors in A'. Now at most [K/8k(K - k)]lAl of the fans F;, F;, ..., FGl contain more than K - k vertices of B'. So F; U F; U - - - U FGl contains a collection of at least [ I - 1/8k - K/8k(K - k)]lAl pairwise-disjoint (k , m)-fans all disjoint from A' and B'; i.e., their endvertices are joined to at least 8k vertices of A'. [The reason is that each fan in the collection has at most K - k vertices in B', and hence at least k paths disjoint from B'. These k paths form a (k , m)-fan.] Since

we have proved (with A' playing the role of A in the theorem) that

We consider next the case where m is even, i.e., m = 2q + 2 where q is a non-negative integer. As in the previous case we prove that

and introduce the fans F,, F,, ..., FmI, the sets SI, S1, ..., Sml; ;nd the fans F ; , F;, ..., FLl. However, we now assume that K = (16k ) +

SUBDIVISIONS OF GRAPHS 27

(16k')' + 8k and we define the fans F;, F;, ... differently as explained below. We consider a collection of disjoint (8k, 1)-fans FY, FY, ..., F; whose centers are in S, U S, U U Sml and whose sets of endvertices S',", S'', ..., Sr are all in A. Furthermore, we assume that p is maximal under these conditions. Clearly,

K / 8 k - 1 S p 6 (A(/8k.

For each endvertex x of FY U F'; U .-- U Fr, we consider the fan F#' (1 =S i S bl) that contains x and a fan 6 ( 1 C j s KWI) containing a neighbor of x in FI and we let P, denote a path in F,! U Fj of length 2 q + 1 from x to a vertex in Sj. The 8kp paths PI are pairwise disjoint and at most p of them contain a center of F'," U FI;' U --. U FF. So F'," U F;" U ... U FF union the paths PI contains a collection of disjoint (4k , 2q + 2)-fans F',', F;, ..., F: where

r 3 p - p / 4 k .

Let the set of endvertices of F; U F ; U ... U F: be denoted B . Note that B S, U S2 U U Sml.

By the maximality of p , each vertex of B is adjacent to at least K - 8k = (16k')' + (16k2), vertices of Sr u Sy U ..- U Sr. So, by Theorem 1, there are subsets A' C Sy U S ! U ... U S'; and B' C B such that

1 r 16kZ 4k'

IB'I S - = -

and each vertex of B\B' has at least 16k' (and hence at least k ) neighbors in A' . Now at most 1B'1/2k of the fans F;, F; , ..., F:' contain more than 2k vertices of B' , so F',' U F ; U ..- u F:contains a collection of at least

r - 1B'1/2k 3 ( 1 - 1/8k2)r

(2k. m)-fans disjoint from B'.

at least At most (A'l/k of these contain more than k vertices of A', so we get

(k. m)-fans which are disjoint from A' u B' and whose ends all have

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more than k neighbors in A'. Since

for k sufficiently large the proof is complete since the existence of g(k , rn) implies the existence of g(k - 1, rn).

Theorem 2 implies the existence of the functionf(k, m) mentioned in the Abstract. Indeed, we have

Theorem 3. f (k , m) =s g(2k, m - 1).

Proof. By Theorem 2 , any graph of minimum degree g(2k, m - 1) contains a set A and a collection of hl pairwise disjoint (2k, m - 1)- fans each endvertex of which has at least 2k neighbors in A . This means that each of these fans can be extended to a (2k, m)-fan with endvertices in A . These fans form a subdivision of a bipartite graph H with 2bI vertices (namely A union the centers of the fans) and 2khI edges. Since H contains a subgraph of minimum degree greater than k (namely any vertex-minimal induced subgraph with at least k times as many edges as vertices) the proof is complete.

As mentioned in the Introduction, the graph H in the Abstract cannot be chosen to be complete. But maybe the following holds.

Problem. Does there exist a function h(k) (k a natural number) such that each graph of minimum degree h(k) contains a subdivision of the complete graph of order k such that each edge is subdivided the same number of times?

References

[ I ] B. BollobAs, Cycles modulo k. Bull. London Math. Soc. 9 (1977) 97- 98.

[21 B. Bollobds, Cycles and semi-topological configurations. In Theory and Applications of Graphs. Y . Alavi and D. R. Lick, Eds. Springer Lecture Notes in Math. 642, Springer, New York (1978) 66-74.

[3] W. Mader, Existenz gewisser Konfigurationen in n-gesattigten Graphen und in Graphen genugend grosser Kantendichte. Math. Ann. 194

141 C. Thomassen, Graph decomposition with applications to subdivisions and path systems modulo k. J. Graph Theory. 7 (1983) 261-271.

(1971) 295-312.