### The Analytic Reformulation of the Hadwiger Conjecture

#### Abstract

**0.**The

*Hadwiger conjecture*[recall that the famous four-color problem is a special case of the Hadwiger conjecture] states that every graph $G$ satisfies $\chi(G) \leq \eta(G)$ [where $\chi(G)$ is the

*chromatic number of $G$*, and $\eta(G)$ is the

*hadwiger number*of $G$ (i.e.\ the maximum of $p$ such that $G$ is

*contractible*to the complete graph $K_{p}$)]. In this paper, via an original speech and simple results, we rigorously simplify the understanding of the Hadwiger conjecture. It will appear that to solve the famous Hadwiger conjecture is equivalent to solve an analytic conjecture stated on a very small class of graphs.

**\textbf{1.}**The

*Hadwiger conjecture*(see [3] or [4] or [6] or [7] or [8] or [9] or [10]) and the

*Berge problem*(see [2] or [4] or [5] or [6] or [8]) are well known. Recall in a graph $G=[V(G),E(G),\chi(G),\omega(G), \bar{G}]$, $V(G)$ is the set of vertices, $E(G)$ is the set of edges, $\chi(G)$ is the chromatic number, $\omega(G)$ is the clique number and $\bar{G}$ is the complementary graph of $G$. The

*Hadwiger conjecture*[recall that the famous four-color problem is a special case of the Hadwiger conjecture] states that every graph $G$ satisfies $\chi(G) \leq \eta(G)$ [where $\eta(G)$ is the hadwiger number of $G$ (i.e.\ the maximum of $p$ such that $G$ is

*contractible*to the complete graph $K_{p}$)]. We say that a graph $B$ is

*berge*if every $B'\in \lbrace B, \bar{B}\rbrace$ does not contain an induced cycle of odd length $\geq 5$. A graph $G$ is

*perfect*if every induced subgraph $G'$ of $G$ satisfies $\chi(G')=\omega(G')$. Indeed, the

*Berge problem*(see [4] or [5] or [6]) consists to show that $\chi(B)=\omega(B)$ for every

*berge*graph $B$ [we recall (see {\bf{[0]), that the

*Berge problem*was solved in a paper of $146$ pages long by Chudnovsky, Robertson, Seymour and Thomas]. In [4], it is presented an original investigation around the

*Berge problem*and the

*Hadwiger conjecture*, and, via two simple Theorems, it is shown that the

*Berge problem*and the

*Hadwiger conjecture*are curiously resembling, so resembling that they seem identical [indeed, they can be restated in ways that resemble each other (see [4])].

*Now, in this paper, via only original speech and results, we rigorously simplify the understanding of the Hadwiger conjecture. Moreover, it will appear that to solve the Hadwiger conjecture is equivalent to solve an analytic conjecture stated on a very small class of graphs*

#### Keywords

True pal; Hadwiger index; Parent; Optimal coloration; Uniform graph; Relative subgraph; Hadwigerian; Hadwigerian subgraph; Maximal hadwigerian subgraph; Hadwiger caliber

#### Full Text:

PDFDOI: http://dx.doi.org/10.26713%2Fcma.v2i1.131

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