Related papers: Disproof of the Odd Hadwiger Conjecture

We prove that $\{\overline{K_3}, H\}$-free graphs are not counterexamples to Hadwiger's Conjecture, where $H$ is any one of 33 graphs on seven, eight, or nine vertices, or $H=K_8$. This improves on past results of Plummer-Stiebitz-Toft,…

Archdeacon (1987) proved that graphs embeddable on a fixed surface can be $3$-coloured so that each colour class induces a subgraph of bounded maximum degree. Edwards, Kang, Kim, Oum and Seymour (2015) proved that graphs with no…

The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the smallest $k$ for which it admits a $k$-coloring where every color class induces an acyclic subgraph. Inspired by Hadwiger's conjecture for undirected graphs, several groups of…

Hadwiger's conjecture asserts that any graph contains a clique minor with order no less than the chromatic number of the graph. We prove that this well-known conjecture is true for all graphs if and only if it is true for squares of split…

Hadwiger's conjecture asserts that if a simple graph $G$ has no $K_{t+1}$ minor, then its vertex set $V(G)$ can be partitioned into $t$ stable sets. This is still open, but we prove under the same hypotheses that $V(G)$ can be partitioned…

Given a graph $H$, let us denote by $f_\chi(H)$ and $f_\ell(H)$, respectively, the maximum chromatic number and the maximum list chromatic number of $H$-minor-free graphs. Hadwiger's famous coloring conjecture from 1943 states that…

We propose local versions of Hadwiger's Conjecture, where only balls of radius $\Omega(\log(v(G)))$ around each vertex are required to be $K_{t}$-minor-free. We ask: if a graph is locally-$K_{t}$-minor-free, is it $t$-colourable? We show…

In 2001, Woodall conjectured that for every pair of integers $s,t \ge 1$, all graphs without a $K_{s,t}$-minor are $(s+t-1)$-choosable. In this note we refute this conjecture in a strong form: We prove that for every choice of constants…

It is well known that for any integers $k$ and $g$, there is a graph with chromatic number at least $k$ and girth at least $g$. In 1960's, Erd\H{o}s and Hajnal conjectured that for any $k$ and $g$, there exists a number $h(k,g)$, such that…

We narrow the gap between the family of graphs that do and the family of graphs that do not satisfy the fat minor conjecture by obtaining much simpler counterexamples than were previously known, including $K_t, t \geq 6$ and $K_{s,t}, s,t…

We present an infinite family of 3-connected non-bipartite graphs with chromatic roots in the interval (1,2) thus resolving a conjecture of Jackson's in the negative. In addition, we briefly consider other graph classes that are conjectured…

Scott proved in 1997 that for any tree $T$, every graph with bounded clique number which does not contain any subdivision of $T$ as an induced subgraph has bounded chromatic number. Scott also conjectured that the same should hold if $T$ is…

An odd graph is a finite graph all of whose vertices have odd degrees. Given graph $G$ is decomposable into $k$ odd subgraphs if its edge set can be partitioned into $k$ subsets each of which induces an odd subgraph of $G$. The minimum…

Let $K_7^{\vee}$ denote the graph obtained from the complete graph on seven vertices by deleting two edges with a common end. Motivated by Hadwiger's conjecture, we prove that every graph with no $K_7^{\vee}$-minor is $6$-colorable.

The odd chromatic number and the conflict-free chromatic number are new graph parameters introduced by Petru\v{s}evski and \v{S}krekovski [2021] and Fabrici, Lu\v{z}ar, Rindo\v{s}ov\'a and Sot\'ak [2022] respectively. In this note, we show…

The defective chromatic number of a graph class is the infimum $k$ such that there exists an integer $d$ such that every graph in this class can be partitioned into at most $k$ induced subgraphs with maximum degree at most $d$. Finding the…

Let $f_{o}(G)$ be the maximum order of an odd induced subgraph of $G$. In 1992, Scott proposed a conjecture that $f_{o}(G)\geq \frac {n} {2\chi(G)}$ for a graph $G$ of order $n$ without isolated vertices, where $\chi(G)$ is the chromatic…

We investigate the effect of a fixed forbidden clique minor upon the strong chromatic index, both in multigraphs and in simple graphs. We conjecture for each $k\ge 4$ that any $K_k$-minor-free multigraph of maximum degree $\Delta$ has…

A connected graph $G$ with chromatic number $t$ is double-critical if $G \backslash \{x, y\}$ is $(t - 2)$-colorable for each edge $xy \in E(G)$. The complete graphs are the only known examples of double-critical graphs. A long-standing…

In this paper, we prove a generalization of a conjecture of Erd\"{o}s, about the chromatic number of certain Kneser-type hypergraphs. For integers $n,k,r,s$ with $n\ge rk$ and $2\le s\le r$, the $r$-uniform general Kneser hypergraph…