On topological relaxations of chromatic conjectures

There are several famous unsolved conjectures about the chromatic number that were relaxed and already proven to hold for the fractional chromatic number. We discuss similar relaxations for the topological lower bound(s) of the chromatic number. In particular, we prove that such a relaxed version is true for the Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of Hadwiger from this point of view. For the latter, a similar statement was already proven in an earlier paper of the first author with G. Tardos, our main concern here is that the so-called odd Hadwiger conjecture looks much more difficult in this respect. We prove that the statement of the odd Hadwiger conjecture holds for large enough Kneser graphs and Schrijver graphs of any fixed chromatic number

Fractional total colourings of graphs of high girth

Reed conjectured that for every epsilon>0 and Delta there exists g such that the fractional total chromatic number of a graph with maximum degree Delta and girth at least g is at most Delta+1+epsilon. We prove the conjecture for Delta=3 and for even Delta>=4 in the following stronger form: For each of these values of Delta, there exists g such that the fractional total chromatic number of any graph with maximum degree Delta and girth at least g is equal to Delta+1

Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)

We survey work on coloring, list coloring, and painting squares of graphs; in particular, we consider strong edge-coloring. We focus primarily on planar graphs and other sparse classes of graphs.Comment: 32 pages, 13 figures and tables, plus 195-entry bibliography, comments are welcome, published as a Dynamic Survey in Electronic Journal of Combinatoric

Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures

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 $\omega, \Delta, \chi$ 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 $v$ depends on its degree, denoted $d(v)$, and the size of the largest clique containing it, denoted $\omega(v)$. 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 $L$-colorable if $L$ is a list-assignment such that $$|L(v)| \geq \lceil (1 - \varepsilon)(d(v) + 1) + \varepsilon\omega(v))\rceil$$ for each vertex $v$ and $\varepsilon \leq 1/2$, and we prove this for $\varepsilon \leq 1/330$ under some mild additional assumptions. We also conjecture the ``$\mathrm{mad}$ version'' of Reed's Conjecture, even for list-coloring. That is, for $\varepsilon \leq 1/2$, every graph $G$ satisfies \chi_\ell(G) \leq \lceil (1 - \varepsilon)(\mad(G) + 1) + \varepsilon\omega(G)\rceil, where $\mathrm{mad}(G)$ is the maximum average degree of $G$. We prove this conjecture for small values of $\varepsilon$, assuming $\omega(G) \leq \mathrm{mad}(G) - \log^{10}\mathrm{mad}(G)$. 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 $K_t$-minor free graphs by a constant factor. We provide a unified treatment of coloring graphs with small clique number. We prove that for $\Delta$ sufficiently large, if $G$ is a graph of maximum degree at most $\Delta$ with list-assignment $L$ such that for each vertex $v\in V(G)$, $$|L(v)| \geq 72\cdot d(v)\min\left\{\sqrt{\frac{\ln(\omega(v))}{\ln(d(v))}}, \frac{\omega(v)\ln(\ln(d(v)))}{\ln(d(v))}, \frac{\log_2(\chi(G[N(v)]) + 1)}{\ln(d(v))}\right\}$$ and $d(v) \geq \ln^2\Delta$, then $G$ is $L$-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 $G$ with $\omega(G)\leq \omega$ and $\Delta(G)\leq \Delta$ for $\Delta$ sufficiently large: $$\chi(G) \leq 72\Delta\sqrt{\frac{\ln\omega}{\ln\Delta}}.$$ 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 $[0, 1]$-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 $\omega, \Delta, \chi$ 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 $G$ is a graph and $f : V(G) \rightarrow [0, 1]$ such that every clique $K$ in $G$ satisfies $\sum_{v\in K}f(v) \leq 1$ and every vertex $v\in V(G)$ demands $f(v) \leq 1/(d(v) + 1/2)$, then $G$ has a fractional coloring $\phi$ in which the measure of $\phi(v)$ for each vertex $v\in V(G)$ is at least $f(v)$. This result generalizes the Caro-Wei Theorem and improves its bound on the independence number, and it is tight for the 5-cycle