Coloring triangle‐free graphs with local list sizes

Abstract

We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow‐up work of Bernshteyn) on the (list) chromatic number of triangle‐free graphs. In both our results, we permit the amount of color made available to vertices of lower degree to be accordingly lower. One result concerns list coloring and correspondence coloring, while the other concerns fractional coloring. Our proof of the second illustrates the use of the hard‐core model to prove a Johansson‐type result, which may be of independent interest.

Occupancy fraction, fractional colouring, and triangle fraction

“…In Section 5, we give some indication that our application of this analysis is essentially tight. The same method was used for similar results specific to triangle-free graphs [8,10]; to an extent, the present work generalises that earlier work.…”

“…is a lower bound on the size of a largest independent set in G. We have presented in an earlier work [8,Lemma 3] how to construct a fractional colouring of a graph G using a probability distribution over the independent sets of any induced subgraph H of G. Informally speaking, if this probability distribution has the property that for every vertex ∈ v H, either v has a good chance of belonging to the corresponding random independent set I, or the expected number of neighbours of v in I is large, then we obtain an upper bound on χ G ( )…”

“…This is done using a greedy fractional colouring algorithm, and we refer the reader to [8] or [14,Section 21.3] for a description of this algorithm. Note that the statement in [8] is significantly more general than what we need here: to obtain the statement below we take r = 1, rename α α , 1 2 to α β , , and trade the colour-measure definition of fractional colouring for the conceptually simpler weighted independent set version. The conclusion of [8,Lemma 3] gives what we want here as when each vertex v is fractionally coloured with an interval of the…”

Bipartite Induced Density in Triangle-Free Graphs

“…The same subset of the present authors with Davies [5] have found a surprisingly short proof for the fractional colouring analogue of Theorem 4.5, via elementary properties of the hard-core model. Using this, there is a streamlined, self-contained proof (of about two pages overall) of Theorem 1.4.…”

Separation Choosability and Dense Bipartite Induced Subgraphs

“…Since the posting of our manuscript to arXiv, the second half of our paper has precipitated a number of significant further developments. [13] has improved the asymptotic leading constant to 2 in Theorem 3.5.…”