Related papers: Counterexamples to Hedetniemi's conjecture

Hedetniemi conjectured in 1966 that $\chi(G \times H) = \min\{\chi(G), \chi(H)\}$ for all graphs $G$ and $H$. Here $G\times H$ is the graph with vertex set $ V(G)\times V(H)$ defined by putting $(x,y)$ and $(x',y')$ adjacent if and only if…

Extending a recent breakthrough of Shitov, we prove that the chromatic number of the tensor product of two graphs can be a constant factor smaller than the minimum chromatic number of the two graphs. More precisely, we prove that there…

For a graph $G$, let $\chi (G)$ denote the chromatic number. In graph theory, the following famous conjecture posed by Hedetniemi has been studied: For two graphs $G$ and $H$, $\chi (G\times H)=\min\{\chi (G),\chi (H)\}$, where $G \times H$…

A $50$ years unsolved conjecture by Hedetniemi [{\it Homomorphisms of graphs and automata, \newblock {\em Thesis (Ph.D.)--University of Michigan}, 1966}] asserts that the chromatic number of the categorical product of two graphs $G$ and $H$…

We construct a connected graph H such that (1) \chi(H) = \omega; (2) K_\omega, the complete graph on \omega points, is not a minor of H. Therefore Hadwiger's conjecture does not hold for graphs with infinite coloring number.

In 1966, Hedetniemi conjectured that for any positive integer $n$ and graphs $G$ and $H$, if neither $G$ nor $H$ is $n$-colourable, then $G \times H$ is not $n$-colourable. This conjecture has received significant attention over the past…

It is proved that in Godel's constructible universe, for every infinite successor cardinal k, there exist graphs G and H of size and chromatic number k, for which the tensor product graph (G x H) is countably chromatic.

We prove that $\min\{\chi(G), \chi(H)\} - \chi(G\times H)$ can be arbitrarily large, and that if Stahl's conjecture on the multichromatic number of Kneser graphs holds, then $\min\{\chi(G), \chi(H)\}/\chi(G\times H) \leq 1/2 + \epsilon$ for…

One of the most famous conjecture in graph theory is Hedetniemi's conjecture stating that the chromatic number of the categorical product of graphs is the minimum of their chromatic numbers. Using a suitable extension of the definition of…

Hedetniemi's conjecture~\cite{hedetniemi1966homomorphisms} for $c$-colorings states that the tensor product $G \times H$ is $c$-colorable if and only if $G$ or $H$ is $c$-colorable. El-Zahar and Sauer~\cite{El-ZaharS85} proved it for $c =…

We introduce the following weak version of Hadwiger's conjecture: If $G$ is a graph and $\kappa$ is a cardinal such that there is no coloring map $c:G \to \kappa$, then $K_\kappa$ is a minor of $G$. We prove that this statement is true for…

We start by building up some theory to state Wagner's Theorem, and then prove it using Kuratowski's Theorem, a proof of which is found in Diester (2000). Following this, we establish some connections between the chromatic number of a graph…

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…

Let $h(G)$ denote the largest $t$ such that $G$ contains $K_t$ as a minor and $\chi(G)$ be the chromatic number of $G$ respectively. In 1943, Hadwiger conjectured that $h(G) \geq \chi(G)$ for any graph $G$. In this paper, we prove that…

The $\mathbb{Z}_2$-index ${\rm ind}(X)$ of a $\mathbb{Z}_2$-CW-complex $X$ is the smallest number $n$ such that there is a $\mathbb{Z}_2$-map from $X$ to $S^n$. Here we consider $S^n$ as a $\mathbb{Z}_2$-space by the antipodal map.…

The first non-obvious case of Hadwiger's Conjecture states that every graph $G$ with chromatic number at least 4 has a $K_4$ minor. We give a new proof that derives the $K_4$ minor from a proper 3-coloring of a subgraph of $G$.

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…

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…

The Odd Hadwiger's conjecture, formulated by Gerards and Seymour in 1995, is a substantial strengthening of Hadwiger's famous coloring conjecture from 1943. We investigate whether the hierarchy of topological lower bounds on the chromatic…

Let $\chi(G)$ denote the chromatic number of a graph and $\chi_v(G)$ denote the vector chromatic number. For all graphs $\chi_v(G) \le \chi(G)$ and for some graphs $\chi_v(G) \ll \chi(G)$. Galtman proved that Hoffman's well-known lower…