Related papers: Hadwiger's conjecture for graphs with infinite chr…

We prove that the existence of a non-special tree of size $\lambda$ is equivalent to the existence of an uncountably chromatic graph with no $K_{\omega_1}$ minor of size $\lambda$, establishing a connection between the special tree number…

The Hadwiger number of a graph $G$, denoted $h(G)$, is the largest integer $t$ such that $G$ contains $K_t$ as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph $G$, $h(G) \ge \chi(G)$, where $\chi(G)$ denotes…

For a graph G, let h(G) denote the largest k such that G has k pairwise disjoint pairwise adjacent connected nonempty subgraphs, and let s(G) denote the largest k such that G has k pairwise disjoint pairwise adjacent connected subgraphs of…

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…

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…

Hadwiger's Conjecture from 1943 states that every graph with no $K_{t}$ minor is $(t-1)$-colorable; it remains wide open for all $t\ge 7$. For positive integers $t$ and $s$, let $\mathcal{K}_t^{-s}$ denote the family of graphs obtained from…

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 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…

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 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…

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…

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…

Hadwiger's Conjecture asserts that every $K_t$-minor-free graph has a proper $(t-1)$-colouring. We relax the conclusion in Hadwiger's Conjecture via improper colourings. We prove that every $K_t$-minor-free graph is $(2t-2)$-colourable with…

Hadwiger's conjecture claims that any graph with no $K_t$ minor is $(t - 1)$-colorable. This has been proved for $t \le 6$, but remains open for $t \ge 7$. As a variant of this conjecture, graphs with no $K_t^=$ minor have been considered,…

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…

We propose an algorithm to reduce a k-chromatic graph to a complete graph of largest possible order through a well defined sequence of contractions. We introduce a new matrix called transparency matrix and state its properties. We then…

The classical Hadwiger conjecture dating back to 1940's states that any graph of chromatic number at least $r$ has the clique of order $r$ as a minor. Hadwiger's conjecture is an example of a well studied class of problems asking how large…

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…

Motivated by the famous Hadwiger's Conjecture, we study the properties of $8$-contraction-critical graphs with no $K_7$ minor; we prove that every $8$-contraction-critical graph with no $K_7$ minor has at most one vertex of degree $8$,…

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 $G \times H$…