Disjoint subgraphs of large maximum degree

Abstract

Erdős [in: Chartrand (Ed.), The Theory and Applications of Graphs, Wiley, New York, 1981, p. 331] conjectured that the vertices of any graph with fewer than 2n+1 2 − n 2 edges can be split into two parts, both parts inducing subgraphs of maximum degree less than n. Recently, the first named author [Combinatorica 21 (2001) 403–412] disproved this conjecture. In this paper we consider further questions arising out of the conjecture. First of all, we give couterexamples to the conjecture having only 2n+80 vertices for large n. (The above counterexample had around n 3/2/ 2 vertices, though it had many fewer edges than our examples.) We also define the function b(n,m) to be the minimum size of a graph G such that, for any partition V(G)=A ∪ B , either Δ(G[A])⩾n or Δ(G[B])⩾m holds. In this terminology, Erdős's conjecture was b(n,n)= 2n+1 2 − n 2 . We prove that b(n,m)=2nm−m 2+ O( m )n for n⩾m,b(n,1)=4n−2 for n⩾7, and b(n,2)=6n+ O(1) . Let m(n,k,j) be the minimum size of a graph G on n+k vertices in which Δ(G[A])⩾n for every (n+j)-set A⊂V(G). We prove that, if k= o(n(n+j)/ log n) , then m(n,k,j)=(1+ o(1)) 1+ k−j 2n+2j (k−j+1)n as n → ∞ . The upper bound here disproves a conjecture made by Erdős, Reid, Schelp and Staton [Discrete Math. 158 (1996) 283–286].