Separation Choosability and Dense Bipartite Induced Subgraphs

Kang, Ross J.; Thomassé, Stéphan

doi:10.1017/s0963548319000026

Cited by 16 publications

(35 citation statements)

References 28 publications

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“…As two other pieces of evidence for Conjecture 6.2, we note that [2] showed that a triangle-free, ddegenerate graph may have χ(G) as large as d; the graph G which achieves this indeed has χ f (G) d log d . Finally, we note that [6] has found other applications of Conjecture 6.2, which would cleanly show certain other bounds regarding induced bipartite graphs. Slightly weaker versions of these bounds have been found by other more laborious methods.…”

Section: Conjectured Tight Boundsmentioning

confidence: 54%

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Some Results on Chromatic Number as a Function of Triangle Count

Harris¹

2019

SIAM J. Discrete Math.

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A variety of powerful extremal results have been shown for the chromatic number of triangle-free graphs. Three noteworthy bounds are in terms of the number of vertices, edges, and maximum degree given by Poljak & Tuza (1994), and Johansson. There have been comparatively fewer works extending these types of bounds to graphs with a small number of triangles. One noteworthy exception is a result of Alon et. al (1999) bounding the chromatic number for graphs with low degree and few triangles per vertex; this bound is nearly the same as for triangle-free graphs. This type of parametrization is much less rigid, and has appeared in dozens of combinatorial constructions.In this paper, we show a similar type of result for χ(G) as a function of the number of vertices n, the number of edges m, as well as the triangle count (both local and global measures). Our results smoothly interpolate between the generic bounds true for all graphs and bounds for triangle-free graphs. Our results are tight for most of these cases; we show how an open problem regarding fractional chromatic number and degeneracy in triangle-free graphs can resolve the small remaining gap in our bounds.

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Section: Conjectured Tight Boundsmentioning

confidence: 54%

“…We want to show that χ(G) ≤ f (n, t); thus, it suffices to show that s √ n + 6 1/3 ys 3t 2/3 ≥ 1 (6) Substituting in the value for s, simple algebraic manipulations show that this is equivalent to showing:…”

Section: Proofmentioning

confidence: 99%

Some Results on Chromatic Number as a Function of Triangle Count

Harris¹

2019

SIAM J. Discrete Math.

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“…In this subsection we prove that Conjecture 1.1 holds for graphs with minimum degree at least n Ω (1) , which also proves the lower bound in Theorem 1.3 (iii). The following lemma appeared as [16,Theorem 3.4], and is a corollary of Johansson's theorem [22] for colouring triangle-free graphs and a connection between the fractional chromatic number and dense bipartite induced subgraphs (see [16,Theorem 3.1] and a discussion in Section 6).…”

Section: Graphs With Polynomially-large Minimum Degreementioning

confidence: 99%

“…In addition to maximising the number of edges, it is also very natural to study induced bipartite subgraphs of large minimum degree. This more local approach was taken recently by Esperet, Kang and Thomassé in [16], where they made a number of intriguing conjectures. Conjecture 1.1.…”

Section: Introductionmentioning

confidence: 99%

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Dense Induced Bipartite Subgraphs in Triangle-Free Graphs

et al. 2020

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The problem of finding dense induced bipartite subgraphs in H-free graphs has a long history, and was posed 30 years ago by Erdős, Faudree, Pach and Spencer. In this paper, we obtain several results in this direction. First we prove that any H-free graph with minimum degree at least d contains an induced bipartite subgraph of minimum degree at least c H log d/ log log d, thus nearly confirming one and proving another conjecture of Esperet, Kang and Thomassé. Complementing this result, we further obtain optimal bounds for this problem in the case of dense triangle-free graphs, and we also answer a question of Erdős, Janson,

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Single‐conflict colouring

Dvořák

Kang

Ozeki

2020

Journal of Graph Theory

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Given a multigraph, suppose that each vertex is given a local assignment of k colours to its incident edges. We are interested in whether there is a choice of one local colour per vertex such that no edge has both of its local colours chosen. The least k for which this is always possible given any set of local assignments we call the single‐conflict chromatic number of the graph. This parameter is closely related to separation choosability and adaptable choosability. We show that single‐conflict chromatic number of simple graphs embeddable on a surface of Euler genus g is OMathClass-open(g1∕4log gMathClass-close) as g→∞. This is sharp up to the logarithmic factor.

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Separation Choosability and Dense Bipartite Induced Subgraphs

Cited by 16 publications

References 28 publications

Some Results on Chromatic Number as a Function of Triangle Count

Some Results on Chromatic Number as a Function of Triangle Count

Dense Induced Bipartite Subgraphs in Triangle-Free Graphs

Single‐conflict colouring

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