On reducible partition of graphs and its application to Hadwiger conjecture

TL;DR

This paper introduces reducible and special reducible partitions of graphs to analyze the Hadwiger conjecture, providing new tools for understanding graph coloring related to minors.

Contribution

It proposes the concepts of reducible and special reducible partitions, demonstrating their existence for any graph and applying them to study $K_{n+1}$-free graphs in relation to the Hadwiger conjecture.

Findings

Existence of reducible and special reducible partitions for all graphs.

Application of SRP to derive conclusions on graph coloring.

Potential implications for the Hadwiger conjecture.

Abstract

An undirected graph $H$ is called a minor of the graph $G$ if $H$ can be formed from $G$ by deleting edges and vertices and by contracting edges. If $G$ does not have a graph $H$ as a minor, then we say that $G$ is $H$-free. Hadwiger conjecture claim that the chromatic number of $G$ may be closely related to whether it contains $K_{n+1}$ minors. To study the coloring of a $K_{n+1}$-free $G$, we propose a new concept of reducible partition of vertex set $V_G$ of $G$. A reducible partition(RP) of a graph $G$ with $K_n$ minors and without $K_{n+1}$ minors is defined as a two-tuples $\{S_1 \subseteq V_G,S_2\subseteq V_G\}$ which satisfy the following condisions:\\ (1) $S_1 \cup S_2 = V_G, S_1 \cap S_2 = \emptyset $\\ (2) $S_2$ is dominated by $S_1$, \\ (3) the induced subgraph $G\left[S_1\right]$ is a forest,\\ (4) the induced subgraph $G\left[S_2\right]$ is $K_{n}$-free.\\ Further, one can obtain a special reducible partition(SRP) $\{S_1,S_2\}$ of $V_G$, which satisf the following condisions:\\ (1) $S_1 \cup S_2 = V_G, S_1 \cap S_2 = \emptyset $ \\ (2) $S_1$ is an independent set,\\ (4) the induced subgraph $G\left[S_2\right]$ is $K_{n}$-free.\\ We will show that both SRP and RP are always exist for any graph. With the SRP of a $K_{n+1}$-free graph $G$, one can obtain some usefull conclusion on the coloring of $G$.