Maximum k‐colorable subgraphs

Abstract

A lower bound is established on the number of edges in a maximum kcolorable subgraph of a loopless graph G. For the special case of 3-regular graphs, lower bounds are also determined on the maximum number of edges in a bipartite subgraph whose color classes are of equal size.

Judicious partitions and related problems

“…A similar idea (in this case, contracting colour classes to a single vertex) gives the following simple bound observed by several authors [3,6,32,34]. …”

“…Note that if G is k-regular then any partition V (G) = V 1 ∪ V 2 with |V 1 | = |V 2 | has e(V 1 ) = e(V 2 ), so Max Bisection gives a bound on g(G). For instance, for k = 3, Locke [34] showed that every cubic K 4 -free graph has a bisection of size at least A. Scott 11e(G)/15; it follows that g(G) ≤ 2e(G)/15, which is better than the bound e(G)/6 from Theorem 3.12.…”

“…For graphs with larger girth, it should be possible to get stronger results (see [13,19,29,34,46]). One approach here is to combine a good colouring with Theorem 2.2.…”

Exact Bounds for Judicious Partitions of Graphs

“…(In fact, simple proofs can be read out of Lehel and Tuza [11] and Locke [12].) Alon [1] proved that there is some c > 0 such that if m/2 is a sufficiently large square then we can improve on (1) by cm 1/4 , while it is never possible to improve on (1) by more that O(m 1/4 ).…”

Maximum cuts of graphs with forbidden cycles

“…Edwards [10,11] proved that for every m f (m) ≥ m 2 + 1 4 2m 1) and noticed that this is tight when m = k 2 for odd integers k. For more information on f (m) and some related topics, we refer the reader to [1,3,5,6,8,14,15,16,21,26,27,28]. For survey articles, see [7,23].…”