SIGNED CLIQUE EDGE-DOMATIC NUMBER OF A GRAPH

Abstract: In this paper, the problem of signed clique edge-domatic number is considered for any finite,nonempty and simple graph G with the maximal clique Kω. At first,using induction method, we study the signed clique edge-domatic number for a class of graphs with maximal cliqueKω(ω = 2, 3). By inducing and summarizing,we obtain that the signed clique edge-domatic number on this class of graphs and the signed clique edge-domatic number of their maximal clique Kω have some relations. Inspired by ways and regulars of the proof of graphs with ω(G) = 2, 3, we take seriously to study the signed clique edge-domatic number for any graph G with Kω(ω ≥ 4) and have proved that the signed clique edge-domatic numbers on G and on Kω(ω ≥ 4) are equal, so we achieve the signed clique edge-domatic number for any graph G. This conclusion is a practical significance for the promotion of signed edge-domatic number.


Introduction
In this paper, all of the graphs that we consider are finite,nonempty, simple and undirected.Let G = (V (G), E(G)) be a graph with vertex set V (G) and edge set E(G).The order n of G denotes the number of vertices of G and the size m of G denotes the number of edges of G.For any edge e ∈ E(G),N G (e) will denote the open neighborhood of e in G and N G [e] = N G (e) {e} will denote the closed neighborhood.Definition 1. (see [1]) A function f ′ : E(G) → {−1, 1} is called the signed edge dominating function (SEDF) of G if f ′ (e ′ ) ≥ 1,for every e ∈ E(G).The signed edge domination number of G is defined as: Every complete subgraph K ω of G that is not included by other any complete subgraph of G is called a maximal clique of G.The order of maximal complete subgraph is called clique number of G,which is denoted by ω(G).Definition 2. (see [2]) A function f ′ : E(G) → {−1, 1} is called a signed clique edge dominating function(SCEDF)of G if e∈E(Kω) f ′ (e) ≥ 1,for every maximal clique K ω of G. Let F sced (G) is a set of all signed clique edge dominating functions on G.The signed clique edge domination number of G is defined as: If S ⊆ E = E(G), we denote Now we similarly define a signed clique edge-domatic number of G.
i (e) ≤ 1 for each edge e ∈ E(G) be called a signed clique edge dominating family on G. Use Ψ(G) to denote the collection of families of signed clique edge-dominations on G.The signed clique edge-domatic number of G is denoted by V. Lutz, B. Zelinka study the signed domatic number and they give some new ways to solve those problems in [6] and [7], then X.J.Li and J.M.Xu research the signed edge domatic number in [3] and B.Xu proves the signed cycle domination number on graphs in [8].So this article mainly studys the signed clique edge-domatic number.

Basic Properties of the Signed Clique
Edge-domatic Number of a Graph Theorem 1.Let G be a graph of size |E(G)| = m with the signed clique edge domination number γ ′ scl (G) and the signed clique edge-domatic number ,then the definitions imply: Theorem 2. For an arbitrary graph G, d ′ scl (G) is an odd integer.
Proof.For an arbitrary graph G, suppose that be the corresponding signed clique edge dominating family on G.We have for each e ∈ E(G) based on definition.But on the left-hand side of this inequality a sum of an even number of odd summands occurs.Therefore,it is an even number and we obtain d i=1 f ′ i (e) ≤ 0 for each e ∈ E(G).This forces: be the corresponding signed clique edge dominating family on G. Now we have two cases for the size of a maximal clique on G.
This imply: Proof.We can know the value of the signed clique edge domination number on G from its properties(a).We have Corollary 1.For any tree T of order n ≥ 3, then d ′ scl (T ) = 1.
A plane graph G is called a maximal plane graph if its every face is a triangle.
Lemma 2. For any maximal planar graph G of order n(n ≥ 3), then Proof.For any maximal planar graph G of order n(n ≥ 3) has 2n − 4 faces(including its outside boundary) from Lemma 1.
Let F 1 , F 2 , • • • , F 2n−4 be all faces of G and E(F i ) be a set of all edges of the face F i .It is not hard to know that every face Next we will introduce some class of graphs with some properties such that equality can be achieved at above.
Let G 1 = K 3 ,G m is the graph obtained from G m−1 by adding a vertex u and three edges uv 1 , uv 2 , uv 3 ,where v 1 , v 2 , v 3 are three vertices on the same inner face(triangle Clearly,when n = 3, we let G = K 3 ,then G is the maximal clique for itself.Let E(G) = {e 1 , e 2 , e 3 } and f ′ (E(G)) = γ ′ scl (G),there exist exactly values of two edges 1, so we can suppose that f ′ (e 1 ) = 1, Suppose that there exists an SCEDF, say Based on the definition of graph G, we can know that there exists 3-degree vertex v in some inner face of  When n = 3, then Based on the definition of graph G = G n−2 , we can add the vertex u to any inner face F i of G n−3 such that u is adjacent to three vertices of F i .Let V (F i ) = {v 1i , v 2i , v 3i }, so we can get additional three inner faces F 1i , F 2i , F 3i which are formed by {v 1i , v 2i , u},{v 2i , v 3i , u},{v 3i , v 1i , u}.
Lemma 4. For any graph G with size m, let ω(G) = ω(ω ≥ 4) and the number of maximal cliques of G be α(α ≥ 1), We can prove this lemma from two aspects next.
On the other hand, let Theorem 6.For any graph G with size m, let ω(G) = ω(ω ≥ 4) and the number of maximal cliques of G be α(α ≥ 1),then , where all subscripts are taken modulo ω(ω−1)
Case 2.2: k(4k − 1) is even.Define the family of signed clique edge dominating functions f .Note that we have a jump of three in the arguments from f ′ 1 to f ′ 2 and from f ′ for each e ∈ E(K ω ) \ {e k(4k−1)+2 }.
signed edge dominating functions on G with the property that d i=1 f ′ i (e) ≤ 1 for each edge e ∈ E(G) is called a signed edge dominating family on G.The maximum number of functions in a signed edge dominating family on G is the signed edge-domatic number of G, denoted by d ′ s (G).

3
and the values of three edges of H[S] exactly two are 1.Without loss of generality, let g

Proof.
For any graph G with the order n in M , the number of the maximal cliques on G is 2n − 4 by Lemma 1.Let F 1 , F 2 , • • • , F 2n−4 denotes all maximal cliques of G.By theorem1,we determine the signed clique edge-domatic number d ′ scl (G) ≤ 3, since the size of G is |E(G)| = 3n − 6.Therefore, we can get d ′ scl (G) = 1 or 3 by theorem 1 and theorem 2. Now we prove that d ′ scl (G) = 3 is right.Using induction on order n of G, we prove that d ′ scl (G) = 3 holds for all n(n = 3, 4, 5, • • • ).