7 research outputs found
Induced subgraphs of graphs with large chromatic number. VIII. Long odd holes
The final publication is available at Elsevier via https://doi.org/10.1016/j.jctb.2019.05.001 © 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We prove a conjecture of András Gyárfás, that for all k, l, every graph with clique number at most κ and sufficiently large chromatic number has an odd hole of length at least ℓ.Supported by NSF grant DMS-1550991 and US Army Research Office Grant W911NF-16-1-0404. Supported by ONR grant N00014-14-1-0084 and NSF grant DMS-1265563. Current address: Rutgers University, New Brunswick, NJ 08901, USA
Induced subgraphs of graphs with large chromatic number. XII. Distant stars
The Gyarfas-Sumner conjecture asserts that if H is a tree then every graph
with bounded clique number and very large chromatic number contains H as an
induced subgraph. This is still open, although it has been proved for a few
simple families of trees, including trees of radius two, some special trees of
radius three, and subdivided stars. These trees all have the property that
their vertices of degree more than two are clustered quite closely together. In
this paper, we prove the conjecture for two families of trees which do not have
this restriction. As special cases, these families contain all double-ended
brooms and two-legged caterpillars
A survey of -boundedness
If a graph has bounded clique number, and sufficiently large chromatic
number, what can we say about its induced subgraphs? Andr\'as Gy\'arf\'as made
a number of challenging conjectures about this in the early 1980's, which have
remained open until recently; but in the last few years there has been
substantial progress. This is a survey of where we are now
Induced subgraphs of graphs with large chromatic number. IV. Consecutive holes
A hole in a graph is an induced subgraph which is a cycle of length at least
four. We prove that for every positive integer k, every triangle-free graph
with sufficiently large chromatic number contains holes of k consecutive
lengths
Cliques, Degrees, and Coloring: Expanding the ω, Δ, χ paradigm
Many of the most celebrated and influential results in graph coloring, such as Brooks' Theorem and Vizing's Theorem, relate a graph's chromatic number to its clique number or maximum degree. Currently, several of the most important and enticing open problems in coloring, such as Reed's Conjecture, follow this theme.
This thesis both broadens and deepens this classical paradigm.
In Part~1, we tackle list-coloring problems in which the number of colors available to each vertex depends on its degree, denoted , and the size of the largest clique containing it, denoted . We make extensive use of the probabilistic method in this part.
We conjecture the ``list-local version'' of Reed's Conjecture, that is every graph is -colorable if is a list-assignment such that
for each vertex and , and we prove this for under some mild additional assumptions.
We also conjecture the `` version'' of Reed's Conjecture, even for list-coloring. That is, for , every graph satisfies
\chi_\ell(G) \leq \lceil (1 - \varepsilon)(\mad(G) + 1) + \varepsilon\omega(G)\rceil,
where is the maximum average degree of . We prove this conjecture for small values of , assuming . We actually prove a stronger result that improves bounds on the density of critical graphs without large cliques, a long-standing problem, answering a question of Kostochka and Yancey. In the proof, we use a novel application of the discharging method to find a set of vertices for which any precoloring can be extended to the remainder of the graph using the probabilistic method. Our result also makes progress towards Hadwiger's Conjecture: we improve the best known bound on the chromatic number of -minor free graphs by a constant factor.
We provide a unified treatment of coloring graphs with small clique number. We prove that for sufficiently large, if is a graph of maximum degree at most with list-assignment such that for each vertex ,
and , then is -colorable. This result simultaneously implies three famous results of Johansson from the 90s, as well as the following new bound on the chromatic number of any graph with and for sufficiently large:
In Part~2, we introduce and develop the theory of fractional coloring with local demands. A fractional coloring of a graph is an assignment of measurable subsets of the -interval to each vertex such that adjacent vertices receive disjoint sets, and we think of vertices ``demanding'' to receive a set of color of comparatively large measure. We prove and conjecture ``local demands versions'' of various well-known coloring results in the paradigm, including Vizing's Theorem and Molloy's recent breakthrough bound on the chromatic number of triangle-free graphs.
The highlight of this part is the ``local demands version'' of Brooks' Theorem. Namely, we prove that if is a graph and such that every clique in satisfies and every vertex demands , then has a fractional coloring in which the measure of for each vertex is at least . This result generalizes the Caro-Wei Theorem and improves its bound on the independence number, and it is tight for the 5-cycle