Graph Theory

Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem

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