Hadwiger number always upper bounds the chromatic number -- 1852-1943 -- A far-reaching generalisation of Guthrie's postulate

Abstract

In a simple graph $G$, we prove that the \textit{Hadwiger number}, $h(G)$, of the given graph $G$ always upper bounds the \textit{chromatic number}, $\chi(G)$, of the given graph $G$, that is, $\chi(G) \leq h(G)$. This simply stated problem is one of the fundamental questions in combinatorial mathematics, which was made by Hugo Hadwiger in 1943. Consequently, it independently verifies the most famous Four-Color Theorem: the case $h(G) = 4$ is equivalent to the Four-Color Theorem, that is, every planar graph is $4$-colourable. In our novel approach, we use algebraic settings over a finite field $\mathbb{Z}_p$. The algebraic setting, in essence, begins with the complete graph with $h(G)$ vertices (which is a minor, $\mathcal{M}$, of the given graph $G$) and iteratively extends to the simple graph $G$. This conjecture has remained elusive, owing to a lack of understanding of the interdependence, particularly the importance of Lemma 3.1, Lemma 3.2, Lemma 3.3, and Lemma 3.6 in Section 3.