Related papers: Further progress towards Hadwiger's conjecture

Given a graph $H$ and an integer $k\geqslant 2$, let $f_{k}(n,H)$ be the smallest number of colors $C$ such that there exists a proper edge-coloring of the complete graph $K_{n}$ with $C$ colors containing no $k$ vertex-disjoint color…

Given an integer $r\ge1$ and graphs $G, H_1, \ldots, H_r$, we write $G \rightarrow ({H}_1, \ldots, {H}_r)$ if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$ for some $i\in\{1, \ldots, r\}$. A…

We prove that with high probability over the choice of a random graph $G$ from the Erd\H{o}s-R\'enyi distribution $G(n,1/2)$, a natural $n^{O(\varepsilon^2 \log n)}$-time, degree $O(\varepsilon^2 \log n)$ sum-of-squares semidefinite program…

Let $\mathcal{C}$ be a proper minor-closed class of graphs. Given the minors excluded in $\mathcal{C}$, we determine the maximum $q$-centered chromatic number and the maximum $q$th weak coloring number of graphs in $\mathcal{C}$ within an…

Given a graph $G$, a vertex-colouring $\sigma$ of $G$, and a subset $X\subseteq V(G)$, a colour $x \in \sigma(X)$ is said to be \emph{odd} for $X$ in $\sigma$ if it has an odd number of occurrences in $X$. We say that $\sigma$ is an…

A graph H is called common if the total number of copies of H in every graph and its complement asymptotically minimizes for random graphs. A former conjecture of Burr and Rosta, extending a conjecture of Erdos asserted that every graph is…

A $t$-tone $k$-coloring of $G$ assigns to each vertex of $G$ a set of $t$ colors from $\{1,..., k\}$ so that vertices at distance $d$ share fewer than $d$ common colors. The {\it $t$-tone chromatic number} of $G$, denoted $\tau_t(G)$, is…

We prove that for all $\varepsilon>0$, there exists a positive integer $n_0$ such that if $G$ is a graph on $n\geq n_0$ vertices with $\delta(G)\geq\tfrac{1}{2}(1 + \varepsilon)n$, then $G$ satisfies the Total Coloring Conjecture, that is,…

Kamyczura introduced the notion of a majority additive $k$-coloring of a graph $G$ as a function $c: V(G) \to \{1,2,\ldots,k\}$ such that $$\left|\left\{u \in N_G(v):\sum_{w \in N_G(u)} c(w) = s \right\}\right|\leq…

Motivated by different characterizations of planar graphs and the 4-Color Theorem, several structural results concerning graphs of high chromatic number have been obtained. Toward strengthening some of these results, we consider the…

Haj\'os conjectured that every graph containing no subdivision of the complete graph $K_{s+1}$ is properly $s$-colorable. This conjecture was disproved by Catlin. Indeed, the maximum chromatic number of such graphs is $\Omega(s^2/\log s)$.…

As evidence for the Odd Hadwiger Conjecture, Simonyi and Zsb\'an (2010) showed that every Kneser graph $G$ with large enough order (compared to $\chi(G)$) contains a totally odd subdivision of $K_{\chi(G)}$. A recent result of Steiner…

Given $\eta \in [0, 1]$, a colouring $C$ of $V(G)$ is an $\eta$-majority colouring if at most $\eta d^+(v)$ out-neighbours of $v$ have colour $C(v)$, for any $v \in V(G)$. We show that every digraph $G$ equipped with an assignment of lists…

Brooks' theorem states that all connected graphs but odd cycles and cliques can be colored with $\Delta$ colors, where $\Delta$ is the maximum degree of the graph. Such colorings have been shown to admit non-trivial distributed algorithms…

Motivated by Hadwiger's conjecture and related problems for list-coloring, we study graphs $H$ for which every graph with minimum degree at least $|V(H)|-1$ contains $H$ as a minor. We prove that a large class of apex-outerplanar graphs…

A proper $k$-colouring of a graph $G$ is called $h$-conflict-free if every vertex $v$ has at least $\min\, \{h, {\rm deg}(v)\}$ colours appearing exactly once in its neighbourhood. Let $\chi_{\rm pcf}^h(G)$ denote the minimum $k$ such that…

The Hadwiger number h(G) of a graph G is the maximum size of a complete minor of G. Hadwiger's Conjecture states that h(G) >= \chi(G). Since \chi(G) \alpha(G) >= |V(G)|, Hadwiger's Conjecture implies that \alpha(G) h(G) >= |V(G)|. We show…

A strong edge coloring of a graph $G$ is an edge coloring $\phi\,:\,E(G) \rightarrow \mathbb N$ such that each color class forms an induced matching in $G$. The strong chromatic index of $G$, written $\chi'_s(G)$, is the minimum number of…

As an extension of the Brooks theorem, Catlin in 1979 showed that if $H$ is neither an odd cycle nor a complete graph with maximum degree $\Delta(H)$, then $H$ has a vertex $\Delta(H)$-coloring such that one of the color classes is a…

Let $P_k$ be a path, $C_k$ a cycle on $k$ vertices, and $K_{k,k}$ a complete bipartite graph with $k$ vertices on each side of the bipartition. We prove that (1) for any integers $k, t>0$ and a graph $H$ there are finitely many subgraph…