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5-incidence chromatic motif and its application

Xianyong Meng College of Information Science and Engineering, Shandong Agricultural University, Taian, Shandong, 271000, China. E-mail*: [email protected]

Abstract

Incidence coloring of a graph G is a mapping from the set of incidences to a color-set C such that adjacent incidences of G are assigned distinct colors. The incidence coloring conjecture (ICC) states that the incidence coloring number of every graph is at most ∆ + 2, where ∆ is the maximum of a graph. Although ICC is false in general, but it has been showed for any graph with ∆ ≤ 3, ICC holds. It is NP-complete to determine whether a graph with ∆ ≤ 3 is 4-incidence colorable. In this paper, we study some graphs with

χi(G) = 5.

AMS Subject Classification: 05C15 Keywords: Incidence coloring, , incidence coloring number.

1 Introduction

All graphs considered in this paper are undirected, finite and simple graphs, and use standard notations in (see [1]). Let G=(V ,E) be a graph, I(G) = {(v, e)|v ∈ V, e ∈ E, v is incident with e} be the set of incidences of G. We say that two incidences (v, e) and (w, f) are adjacent if and only if one of the following holds: (1)v = w; (2)e = f; and (3) the edge uw equals to e orf. The concept of incidence coloring was first developed by Brualdi and Massey [2] in 1993. An incidence coloring σ of G is a mapping from I(G) to a color-set C such that adjacent incidences of G are assigned distinct colors. If σ : I(G) 7→ C is an incidence coloring of G and k = |C|, then we say that G is k-incidence colorable and σ is called a k-incidence coloring. The minimum cardinality of C for which there exists an incidence coloring σ: I(G) 7−→ C is called the incidence chromatic number of G,

denoted as χi(G).

Biography: Meng xian-yong(1972-), male, native of Taian, Shandong. Lecture of Shandong Agricultural University , engages in the graph theory and probability graph model.

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The incidence coloring number is an important parameter of graph. Frequently, it is very difficult to determine the incidence coloring number of a graph. As showed in [7], it is NP- complete. The incidence coloring conjecture(ICC) states that the incidence coloring number of every graph is at most ∆+2, where ∆ is the maximum degree of a graph. Although ICC is false in general, it has been showed that for some graphs, in particular graphs with ∆ ≤ 3 , the ICC holds. The incidence chromatic number of some special classes of graphs has been determined, for details see literature [3][4][5][7]. But, for some graphs, it is still needed to study whether its incidence chromatic number is ∆ + 1 or ∆ + 2. In this paper, we study some graphs with

χi(G) = 5.

2 5-incidence chromatic motif .

In this section, we first study a particular graph with the incidence coloring number is 5. Then on it, we determine the incidence coloring number of some graphs. A k-cycle is called a l-regular k-cycle, if the degree of the vertices on the cycle are l.A 3-regular 5-cycle is particular case, its structure is very simple(showed in the Fig.1 and Fig.2) , and its incidence coloring number is very easy determined. As showed in the following, for any graph with ∆ ≤ 3, if a 3-regular 5-cycle is its subgraph, then its incidence chromatic number is 5. So a 3-regular 5-cycle can be used as a probe to determine the incidence coloring number of some graphs, whose maximum degree is 3. In this sense, a 3-regular 5-cycle can be called a 5-incidence chromatic motif.

A strong coloring of G is a proper vertex coloring such that for any u, v ∈ NG(v),u and v are assigned distinct colors. If c : V (G) −→ S is a strong vertex coloring of G and |S| = k, then G is called k-strong-vertex colorable and c is a k-strong-vertex coloring of G, where S is a color-set. In this case, we say that G is k-strong-vertex colored. Lemma 2.1[7] Let k be a positive integer. A graph G whose vertices have degree equal to k or 1 is (k+1)-incidence colorable if and only if G is (k + 1)-strong-vertex colorable.. Lemma 2.2[5] Every graph G with ∆ = 3 is 5-incidence colorable.. Theory 2.3 The incidence chromatic number of a 3-regular 5-cycle is 5.

Proof: . We denote a 3-regular 5-cycle by A. Suppose that χi(A) = 4. By above lemma 2.1, A must be 4-vertex-strong-colorable. But A can not be 4-vertex-strong-

colorable when we color it. So χi(A) ≥ 5. According the lemma 2.2, we have χi(G) = 5. s s s s @ @ s s @ s s s@ s

s s s s s s s s

Fig.1 3-regular 5-cycle(without chord). Fig.2 3-regular 5-cycle(with chord)

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Proposition 2.3 For any graph G with ∆(G) = 3, if G has a 3-regular 5-cycle subgraph, then χ(G) = 5.

3 The incidence chromatic number of 3-regular Halin graph .

A Halin-graph is a particular plane graph. For a Halin-graph with δ ≥ 4, it is very easy showed that the incidence coloring number is ∆ + 1[7]. However, for a 3-regular Halin graph, without 3-regualr 5-cycle, we will be difficult to solve the problem about its incidence coloring number. In fact, its incidence coloring number has been solved in [5][8]. Although their proofs seem different, but they all use 3-regula 5-cycle. Now, we can make their proofs more simple.

Definition 1.1 Let G(V,E) be a 3-connected planar graph, f0 be a face without chord on

its boundary (a cycle) and d(v) = 3 for every v ∈ V (f0). When a T, in which all vertices

v ∈ V \ V (f0) satisfies d(v) ≥ 3, is obtained from G(V,E) by deleting all edges on the boundary

of f0, then G(V,E) is called a Halin-graph; The face f0 is called exterior face (the others are

called interior faces ). The every vertex on f0 is called exterior vertex (the others are called interior vertices). Proposition 3.1 Let G be a 3-regular Halin graph. Then G must have 3-regular 5-cycle. S Proof: G is a 3-regular Halin graph, then G =T C, where T is a binary tree and C is a

cycle. Pick an arbitrary leaf of a longest in T to be the root. Take a leaf v1 of the maximal

depth and the other descendant v2 of its parent w. Consider the parent u of w. Either the other 0 0 0 0 descendant w of u is leaf, and v1, v2, w , u, w form a 3-regular 5-cycle, or it has descendants v1, v2, 0 0 0 which are leaves by depth assumption, and v1(v2), v1(v2), w , u, w form a 3-regular 5-cycle. Obviously, we can obtain the following the proposition.

Proposition 2.2 Let G be a 3-regular Halin graph G(G 6= W3). Then χi(G) = 5

4 Remarks

Up to now, the proof used in the problem about the incidence coloring number often gives an specific incidence coloring. In many cases, this is very difficult. But by some particular graph, it may be changed very simple. As above without 3-regular 5-cycle, it is difficult to know whether the incidence coloring number of 3-regular Halin-graphs is 5. As in [7], it is NP-complete to determine if a general graph is k-incidence colorable. So, in the future, we should give more particular graphs(chromatic motifs) to help us determine the incidence coloring number.

References

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[4] M. Maydanskiy, The incidence coloring conjecture for graphs of maximum degree 3. Discrete Math. 292(2005)131-141.

[5] W. C. Shiu, P. K. Sun, Invalid proofs on incidence coloring, Discrete Math.308(2008)6575- 6580

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[8] X. Y. Meng, On the coloring of Pseudo-Halin graph[M], Shandong University of Science and Technology 2004.

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