On reducible partition of graphs and its application to 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 $G$.