Related papers: Hedetniemi's Conjecture Via Altermatic Number

Hadwiger's famous coloring conjecture states that every $t$-chromatic graph contains a $K_t$-minor. Holroyd [Bull. London Math. Soc. 29, (1997), pp. 139--144] conjectured the following strengthening of Hadwiger's conjecture: If $G$ is a…

Let $\phi(k)$ be the minimum number of vertices in a non-$k$-choosable $k$-chromatic graph. The Ohba conjecture, confirmed by Noel, Reed and Wu, asserts that $\phi(k) \ge 2k+2$. This bound is tight if $k$ is even. If $k$ is odd, then it is…

Hadwiger's conjecture states that every $K_t$-minor free graph is $(t-1)$-colorable. A qualitative strengthening of this conjecture raised by Gerards and Seymour, known as the Odd Hadwiger's conjecture, states similarly that every graph…

Combinatorics, in particular graph theory, has a rich history of being a domain of successful applications of tools from other areas of mathematics, including topological methods. Here, we survey the study of the Hom-complexes, and the ways…

Investigating the equality of the chromatic number and the circular chromatic number of graphs has been an active stream of research for last decades. In this regard, Habolhassan and Zhu [Circular chromatic number of Kneser graphs, Journal…

A \emph{domatic} (\emph{total domatic}) \emph{$k$-coloring} of a graph $G$ is an assignment of $k$ colors to the vertices of $G$ such that each vertex contains vertices of all $k$ colors in its closed neighborhood (neighborhood). The…

Given a graph $G$ possibly with multiple edges but no loops, denote by $\Delta$ the {\it maximum degree}, $\mu$ the {\it multiplicity}, $\chi'$ the {\it chromatic index} and $\chi_f'$ the {\it fractional chromatic index} of $G$,…

Hadwiger's Conjecture from 1943 states that every graph with chromatic number $t$ contains a $K_t$ minor. Illingworth and Wood [arXiv:2405.14299] introduced the concept of a ``dominating $K_t$ minor'' and asked whether every graph with…

Pushable homomorphisms and the pushable chromatic number $\chi_p$ of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph $\overrightarrow{G}$, we have…

A subgraph $H$ of a multigraph $G$ is overfull if $ |E(H) | > \Delta(G) \lfloor |V(H)|/2 \rfloor$. Analogous to the Overfull Conjecture proposed by Chetwynd and Hilton in 1986, Stiebitz et al. in 2012 formed the multigraph version of the…

The dichromatic number of a graph $G$ is the maximum integer $k$ such that there exists an orientation of the edges of $G$ such that for every partition of the vertices into fewer than $k$ parts, at least one of the parts must contain a…

The "clustered chromatic number" of a class of graphs is the minimum integer $k$ such that for some integer $c$ every graph in the class is $k$-colourable with monochromatic components of size at most $c$. We prove that for every graph $H$,…

The Hadwiger number mr(G) of a graph G is the largest integer n for which the complete graph K_n on n vertices is a minor of G. Hadwiger conjectured that for every graph G, mr(G) >= chi(G), where chi(G) is the chromatic number of G. In this…

This paper is devoted to studying a topological version of the famous Hedetniemi conjecture which says: The $\mathbb Z/2$-index of the Cartesian product of two $\mathbb Z/2$-spaces is equal to the minimum of their $\mathbb Z/2$-indexes. We…

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…

We study the list chromatic number of Cartesian products of graphs through the Alon-Tarsi number as defined by Jensen and Toft (1995) in their seminal book on graph coloring problems. The Alon-Tarsi number of $G$, $AT(G)$, is the smallest…

Hadwiger's Conjecture states that every $K_{t+1}$-minor-free graph is $t$-colourable. It is widely considered to be one of the most important conjectures in graph theory. If every $K_{t+1}$-minor-free graph has minimum degree at most…

A "clique minor" in a graph G can be thought of as a set of connected subgraphs in G that are pairwise disjoint and pairwise adjacent. The "Hadwiger number" h(G) is the maximum cardinality of a clique minor in G. This paper studies clique…

In this paper, in view of $Z_p$-Tucker lemma, we introduce a lower bound for chromatic number of Kneser hypergraphs which improves Dol'nikov-K{\v{r}}{\'{\i}}{\v{z}} bound. Next, we introduce multiple Kneser hypergraphs and we specify the…

In this paper, we explore algebraic approaches to $d$-improper and $t$-clustered colourings, where the colouring constraints are relaxed to allow some monochromatic edges. Bilu [J. Comb. Theory Ser. B, 96(4):608-613, 2006] proved a…