A NEW VERTEX-COLORING EDGE-WEIGHTING OF COMPLETE GRAPHS

Abstract

Let G = (V ; E) be a simple undirected graph without loops and multiple edges, the vertex and edge sets of it are represented by V = V (G) and E = E(G), respectively. A weighting w of the edges of a graph G induces a coloring of the vertices of G where the color of vertex v, denoted $S_v:={\Sigma}_{e{\ni}v}\;w(e)$. A k-edge-weighting of a graph G is an assignment of an integer weight, w(e) ${\in}${1,2,...,k} to each edge e, such that two vertex-color $S_v$, $S_u$ be distinct for every edge uv. In this paper we determine an exact 3-edge-weighting of complete graphs $k_{3q+1}\;{\forall}_q\;{\in}\;{\mathbb{N}}$. Several open questions are also included.

1. Introduction

Let G = (V ;E) be a simple undirected graph without loops and multiple edges, the vertex and edge sets of it are represented by V = V (G) and E = E(G), respectively. A weighting w of the edges of a graph G induces a coloring of the vertices of G.

A k-edge-weighting of a graph G is an assignment of an integer weight, w(e) ∈ {1,2,...,k} to each edge e. The edge-weighting is proper if for every edge e = uv incident a proper vertex-coloring and the colors of two vertices u, v are distinct, where the color of a vertex v is defined as the sum of the weights on the edges incident to that vertex. Clearly a graph cannot have a k-edge-weighting and vertex-coloring if it has a component which is isomorphic to K2 i.e., an edge component. Throughout this paper, we denoted the color of a vertex v by Sv :=Σu∈V(G) w(uv), such that if vw is not in V(G) w(vw) = 0.

In particular a 3-edge-weighting of G called 1-2-3-edge weighting and vertex coloring of G.

In 2002 [9], Karonski, Luczak and Thomason conjectured that every graph without an edge component permits a 1-2-3-edge weighting and vertex coloring and proved their conjecture for the case of 3-colorable graphs [9]. For k = 2 is not sufficient as seen for instance in complete graphs and cycles of length not divisible by 4.

In 2004, they proved a graph without an edge component permits an 213-edge-weighting and vertex coloring. In continue a constant bound of k = 30 was proved by Addario-Berry, et.al in 2007 [1]. In next year, Addario-Berry’s group improved this bound to k = 16 [2] and also in 2008, T. Wang and Q. Yu improved k to 13 [10]. Recently, its new bounds are k = 5 and k = 6 by Kalkowski, et.al [7,8]. In addition, for further study and more historical details, readers can see the recent papers [3-5].

In this paper, we show that there is a proper 1-2-3-edge weighting and vertex coloring for the complete graphs K3q+1 ∀q ∈ ℕ. and obtain an exact weighting w of all edges e ∈ E(K3q+1) and alternatively a proper color of all vertices v ∈ V (K3q+1). We present these main results in the following theorem:

Theorem 1.1. For the complete graph K3q+1 for every integer number q with the vertex set V (K3q+1) = {v1, v2,...,v3q+1}, there are a 3-edge weighting w : E(K3q+1) → {1, 2, 3} and a vertex-coloring s : V (K3q+1) → {9q, 9q−1,..., 7q+1, 7q, 7q − 2, 7q − 5, 7q − 9,..., 3q + 11, 3q + 7, 3q + 4}.

 

2. Main Results and Algorithm

For a graph complete graph Kn = (V (Kn);E(Kn)), a 3-edge weighting is a function w : E(Kn) → {1, 2, 3} such that is a color for any every vertex v ∈ V (Kn). The edge weighting w implies that E(Kn) = E(Kn)1 ∪ E(Kn)2 ∪ E(Kn)3. Throughout this paper, we denoted the size of edge sets E(Kn), E(Kn)2 and E(Kn)3 by γn, βn and αn, respectively. Obviously

Before proving Theorem 1.1, for a general representation of complete graph K3q+1 ∀q ∈ ℕ, we present a proper 3-edge weighting for all edges incident to a vertex v in 3q + 1 following steps and obtain all summations Sv.

2.1.Algorithm for 1-2-3-edge weighting and vertex coloring of K3q+1(q ≥ 5): At first, we denote all vertices of K3q+1 by v1, v2,...,v3q+1, respectively. Obviously E(K3q+1) = {vivj |i ≠ j, i, j = 1, 2,..., 3q + 1} and this implies that Suppose ∀i = 1, 2,..., 3q + 1;w(vivi) = 0. So, we have

Step(1)- For the vertex v1 label all its edges with 3 (∀vj ∈ V (K3q+1) w(v1vj) = 3 and

Step(2)- For v2 label all v2’s edges with 3, except an edge v2v3q+1, then ∀j = 1, 3, 4,..., 3q, w(v2vj) = 3 and w(v2v3q+1) = 2. Thus S2 = S1−1 = 9q−1.

Step(3)- For v3 label all edges v3vj (j = 1, 2, 4,..., 3q − 1) with 3 and v3v3q, v3v3q+1 with 2. Thus S3 = S2 − 1 = 9q − 2.

Step(4)- For v4 label all edges v4vj (j = 1, 2, 3, 5,..., 3q − 1) with 3, v4v3q with 2 and v4v3q+1 with 1. Thus S4 = S3 − 1 = 9q − 3.

Step(s)- ∀s = 5, 7,..., 2q − 1 label all edges with 3, label with 2 and all edge with 1. Thus

Step(r)- ∀r = 6, 8,..., 2q label all edges with 3, label with 2 and all edge with 1. Thus

Step(2q+1)- For v2q+1, all edges v2q+1vj (j = 1, 2,..., 2q) were labeled with 3. Thus label all edge v2q+1vj (j = 2q + 2,..., 3q + 1) with 1. Thus S2q+1 = 3 × 2q + 2 × 0 + 1 × (q) = 7q = S2q − 1.

Step(2q+2)- For v2q+2, all edges v2q+2vj (j = 1, 2,..., 2q − 2) were labeled with 3, the edge v2q+2v2q−1, v2q+2v2q were labeled with 2 and v2q+2v2q+1 were labeled with 1. Thus label all edges v2q+2vj (j = 2q + 3,..., 3q + 1) with 1 and S2q+2 = 3 × (2q − 2) + 2 × 2 + 1 × (q) = 7q − 2 = S2q+1 − 3.

Step(t)- ∀t = 2q + 3,..., 3q − 2 all edges vtvj (j = 1,..., 6q + 2 − 2t) were labeled with 3, three edges vtv6q+2−2t+i for i = 1, 2, 3 were labeled with 2 and all edges vtvj (j = 6q−2t+6,..., t−1) were labeled with 1. Thus label all edges vtvj (j = t+1,..., 3q+1) with 1 and St = 3×(6q+2−2t)+2×3+1×(2t−3q−5) = 15q + 7 − 4t = St−1 − 4.

Step(3q-1)- v3q−1, v3q−1vj (j = 1, 2, 3, 4) were labeled with 3 and v3q−1v5, v3q−1v6, v3q−1v7 were labeled with 2 and all edge v3q−1vj (j = 8,..., 3q − 2) were labeled with 1. Thus label v3q−1v3q; v3q−1v3q+1 with 1 and S3q−1 = 3×4+2 × 3 + 1 × (3q − 7) = 3q + 11 = S3q−2 − 4.

Step(3q)- For v3q, v3qv1, v3qv2 were labeled with 3 and v3qv3, v3qv4, v3qv5 were labeled with 2 and all edge v3qvj (j = 6,..., 3q − 1) were labeled with 1. Thus label the edge v3q+1v3q with 1 and S3q = 3×2+2×3+1×(3q −5) = 3q +7 = S3q−1 − 4.

Step(3q+1)- Obviously, for the vertex v3q+1, the edge v3q+1v1 were labeled with 3 and v3q+1v2, v3q+1v3 were labeled with 2 and all edge v3q+1vj (j = 4,..., 3q) were labeled with 1. Thus S3q+1 = 3+2×2+1×(3q −3) = 3q +4 = S3q − 3

Now, we start the proof of main theorem as follow.

Proof. Let K3q+1 be a complete graph as order 3q+1 for every integer number q, with the vertex set V (K3q+1) = {v1, v2,..., v3q+1} and the edge set E(K3q+1) = {eij = vivj |vi, vj ∈ V (K3q+1)} (|V (K3q+1)| = 3q+1 and It is easy to see that the above edge-weighting is a nice and proper 3-edge weighting w (or 1-2-3-edge weighting and vertex coloring) of K3q+1 (q ≥ 1). Because and for every edge eij = vivj incident to vi, we have an integer weight w(vivj) ∈ {1, 2, 3} such that this weighting naturally induces two distinct vertex coloring to vertices vi, vj.

An every edge eij = vivj (i, j = 1, 2,..., 3q + 1, i ≥ j) weighted in Step(i) of above 3-edge weighting w. Also, from above 3-edge weighting w, one can see that all vertex color belong to the color set {9q, 9q−1,..., 7q+1, 7q, 7q−2, 7q− 5, 7q − 9,..., 3q + 11, 3q + 7, 3q + 4} (For example, see Figure 1, 2 and 3). In Figure 1, 2 and 3, a proper 1-2-3-edge weighting and vertex coloring of complete graphs K4, K7, K10, K13 and K16 are shown.

FIGURE 1.The 1-2-3-edge weighting and vertex coloring of complete graphs K4, K7 and K10.

FIGURE 2.The 1-2-3-edge weighting and vertex coloring of K13.

FIGURE 3.The 1-2-3-edge weighting and vertex coloring of complete graph K16.

Thus, the 3-edge weighting w : E(K3q+1) → {1, 2, 3} and the vertex coloring s : V (K3q+1) → {9q, 9q −1,..., 7q +1, 7q, 7q −2, 7q −5, 7q −9,..., 3q +11, 3q + 7, 3q + 4} is a proper 1-2-3-edge weighting and vertex coloring of K3q+1 ∀q ∈ ℕ and this complete the proof of theorem.

By using the proof of Theorem 1.1 (the 3-edge weighting w and the vertex coloring s), one can see that the number of all edge weigh 3, 2 and 1 are equal to α3q+1 = 3q2 + 1 = (|E(K3q+1)3|), β3q+1 = 3q − 2 = (|E(K3q+1)2|) and

For example, E(K3q+1)2 = {v2v3q+1, v4v3q, v5v3q−1, v5v3q, v6v3q−1, v7v3q−2, vsv3q−1, v8v3q−2,..., v2q−1v2q+2, v2q−1v2q+3, v2qv2q+2, v2q+2v2q−3, v2q+2v2q−2, v2q+2v2q−1, v2q+3v2q−3, v2q+3v2q−2, v2q+3v2q−1, v2q+4v2q−5, v2q+4v2q−4, v2q+4v2q−3,..., v3q−2v6, v3q−2v7, v3q−2v8, v3q−1v5, v3q−1v6, v3q−1v7, v3qv3v3qv4, v3qv5, v3q+1v2, v3q+1v3}.

 

3. Conclusions and Conjectures

We conclude our paper with the following open questions and conjectures:

Corollary 3.1 (The 1-2-3-conjecture [6,9]). Every connected graph G = (V,E) non-isomorph to K2 (with at least two edges) has an edge labeling f : E → {1, 2, 3} and vertex coloring S : V → {n − 1,...,3n − 3}.

Corollary 3.2 (n vertex coloring). There are distinct numbers of Sv's, v ∈ V (G), of a graph G of order n, for a 1-2-3-edge labeling and vertex coloring.

Corollary 3.3 (Proper vertex coloring). For all graph G of order n, there are the χ(G) numbers of Sv's, v ∈ V (G), with this 1-2-3-edge labeling and vertex Coloring. Where χ(G) is the number colors of the vertices on the graph G.

In this parer, we show that the complete graph K3q+1 (q ≥ 1), recognize in three Conjecture 3.1, 3.2 and 3.3.