On The Hadwiger ’ s Conjecture

The Hadwiger’s conjecture (see [1] or [2] or [3] or [4] or [5] is well known. In this paper, we show an original Theorem which is equivalent to the Hadwiger’s conjecture.


Introduction and prologue
This paper is an original investigation around the Hadwiger conjecture.We recall that in a graph G = [V (G), E(G), χ(G), ω(G)], V (G) is the set of vertices, E(G) is the set of edges, χ(G) is the chromatic number, and ω(G) is the clique number of G.The Hadwiger conjecture states that every graph G is η(G) colorable (i.e.we can color all vertices of G with η(G) colors such that two adjacent vertices do not receive the same color).η(G) is the hadwiger number of G and is the maximum of p such that G is contractible to the complete graph K p ).That being so, this paper is divided into four simple Sections.In Section 2 (Standard definitions), we present briefly some standard definitions known in Graph Theory.In Section 3, we introduce definitions that are not standard, and some elementary properties.In Section 4, we introduce a new graph parameter denoted by τ (τ is called the hadwiger index) and we present elementary properties of this parameter.In Section 5, using the graph parameter τ, we show a simple Theorem which is equivalent to the Hadwiger conjecture.This simple Theorem immediately implies that the Hadwiger conjecture is true if and only if τ(G) = ω(G), for every graph G which is complete ω(G)-partite (τ is the graph parameter defined in Section 4).Here, all results are completely different from all the investigations that have been done around the Hadwiger conjecture in the past.In this paper, every graph is finite, is simple and undirected.

Standard definitions
We start by standard definitions (see [2] or [3] , and two vertices of F are adjacent in F , if and only if they are adjacent in G. is complete; such a subgraph is necessarily an induced subgraph (recall that a graph K is complete if every pair of vertices of K is an edge of K); ω(G) is the size of a largest clique of G, and ω(G) is called the clique number of G.A stable set of a graph G is a set of vertices of G that induces a subgraph with no edges; α(G) is the size of a largest stable set, and α(G) is called the stability number of G.The chromatic number of G (denoted by χ(G)) is the smallest number of colors needed to color all vertices of G such that two adjacent vertices do not receive the same color.It is easy to see: The hadwiger number of a graph G (denoted by η(G)), is the maximum of p such that G is contractible to the complete graph K p .(Recall that, if e is an edge of G incident to x and y, we can obtain a new graph from G by removing the edge e and identifying x and y so that the resulting vertex is incident to all those edges (other than e) originally incident to x or to y.This is called contracting the edge e.If a graph F can be obtained from G by a succession of such edge-contractions, then, G is contractible to F .The maximum of p such that G is contractible to the complete graph K p is the hadwiger number of G, and is denoted by η(G)).The Hadwiger conjecture states that χ(G) ≤ η(G), for every graph G. Clearly we have: Assertion 2.2.Let G and let F be a subgraph of G. Then η(F ) ≤ η(G).

Non-standard definitions and some elementary properties
In this section, we introduce definitions that are not standard.These definitions are determining for our final Theorem.We say that a graph G is a true pal of a graph 2) is also immediate (indeed, let F be graph and let Ξ(F ) = {Y 1 , . . ., Y χ(F ) ] be a partition of V (F ) into χ(F ) stable sets (it is immediate that such a partition Ξ(F ) exists).Now let Q be a graph defined as follows: (i) So, we say that a graph P is a parent of a graph F , if P ∈ Ω trpl(F ).In other words, P is a parent of F , if P is a complete ω(P)-partite graph and P is also a true pal of F (observe that such a P exists, via property (3.1.2) of Assertion 3.1).parent(F ) denotes the set of all parents of a graph F (so, P ∈ parent(F ) means P is a parent of F ). Using the definition of a parent, then the following Assertion is immediate.Assertion 3.2.Let F be a graph and let P ∈ parent(F ).We have the following two properties.

The hadwiger index of a graph
Here, we define the hadwiger index of a graph and a son of a graph, and we also give some elementary properties related to the hadwiger index.We recall (see Section 2) that η(G) is the hadwiger number of G. Using the definition of a true pal (see Section 3), then the following assertion is immediate.
Proof.Properties     Proof.Observe that P ∈ trpl(F ) and apply Proposition 4.3.
We will see in Section 5 that the hadwiger index helps to obtain a simple Theorem which is equivalent to the Hadwiger conjecture.

A simple Theorem which is equivalent to the Hadwiger conjecture
In this section, we prove a simple Theorem which is equivalent to the Hadwiger conjecture.This simple Theorem immediately implies that the Hadwiger conjecture is true if and only if τ(G) = ω(G), for every graph G which is complete ω(G)-partite.We recall (see Introduction or see Section 2) that the Hadwiger

Assertion 3 . 1 .
then H ∈ Ω; . . .etc).Using the definition of Ω, then the following Assertion becomes immediate.Let H ∈ Ω and let F be a graph.Then we have the following two properties.(3.1.1)χ(H) = ω(H).(3.1.2)There exists a graph P ∈ Ω such that P is a true pal of F .Proof.Property (3.1.1)is immediate (use the definition of Ω and note H ∈ Ω).Property (3.1.

Assertion 4 . 1 .Proposition 4 . 2 .
Let G be a graph.Then, there exists a graph S such that G is a true pal of S and η(S) is minimum for this property.Now we define the hadwiger index and a son.Let G be a graph and put(G) = [H; G ∈ trpl(H)]; clearly (G) is the set of all graphs H, such that G is a true pal of H.The hadwiger index of G is denoted by τ(G), where τ(G) = min F ∈ (G)η(F ).In other words, τ(G) = η(F ), where F ∈ (G), and η(F ) is minimum for this property.We say that a graph S is a son of G, if G ∈ trpl(S) and η(S) = τ(G).In other words, a graph S is a son of G, if S ∈ (G) and η(S) = τ(G).In other terms again, a graph S is a son of G, if G is a true pal of S and η(S) is minimum for this property.Observe that such a son exists, via Assertion 4.1.It is immediate that, if S is a son of a graph G, then χ(S) = χ(G) and η(S) ≤ η(G).Let G ∈ Ω.We have the following three properties.

Proposition 4 .
3 clearly says that the hadwiger index τ decreases (In the following sense: G is a true pal of F ⇒ τ(G) ≤ τ(F )).