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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 316501, 6 pages
http://dx.doi.org/10.1155/2013/316501
Research Article

Strong List Edge Coloring of Subcubic Graphs

1School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China
2College of Computer Science and Technology, Zhejiang University of Technology, Hangzhou 310023, China
3Department of Mathematics, West Virginia University, Morgantown, WV 26506–6310, USA

Received 9 February 2013; Accepted 31 March 2013

Academic Editor: Carlo Cattani

Copyright © 2013 Hongping Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study strong list edge coloring of subcubic graphs, and we prove that every subcubic graph with maximum average degree less than 15/7, 27/11, 13/5, and 36/13 can be strongly list edge colored with six, seven, eight, and nine colors, respectively.

1. Introduction

All graphs in this paper are finite and simple. For a graph with vertex set and edge set , a proper edge coloring of is an assignment of colors to the edges of so that no two adjacent edges receive the same color. A strong edge coloring is a proper edge coloring so that two edges adjacent to a common edge receive different colors. The strong chromatic index of , denoted by , is the minimum number of colors needed for a strong edge coloring of .

Strong edge coloring was introduced by Fouquet and Jolivet [1, 2]. This type of coloring can be used to represent the conflict-free channel assignment in radio networks.

Denote by the maximum degree of the graph. In 1985, Erdös and Ne et il conjectured that the strong chromatic index of a graph is at most when is even and when is odd. Andersen proved the conjecture for [3]. Strong edge coloring of cubic Halin graphs has been studied in [4, 5].

Let be the maximum average degree of the graph . Hocquard and Valicov [6] considered the subcubic graphs with bounded maximum average degree, and they proved the following results.

Theorem 1. Let be a subcubic graph.(i)If , then .(ii)If , then .(iii)If , then .(iv)If , then .

The main purpose of this paper is to generalize the study of list version so that the admissible colors on edges are constrained. An edge list of a graph is a mapping that assigns a finite set to each edge of . Denote . We say that is a -edge list if for each edge in . The graph is strongly -edge colorable if there exists a strong edge coloring of such that for every edge of . For a positive integer , a graph is strongly -edge choosable if for every -edge list , is strongly -edge colorable. The strong list chromatic index is the minimum positive integer for which is strongly -edge choosable.

In this paper, we consider strong list edge coloring of subcubic graphs and extend Theorem 1 to the list version. We prove the following theorem.

Theorem 2. Let be a subcubic graph.(i)If , then .(ii)If , then .(iii)If , then .(iv)If , then .

The paper is organized as follows. In Section 2, we will prove two lemmas which will be applied a lot in the proof of Theorem 2. Theorem 2 will be proved in Sections 3, 4, 5, and 6.

Before proceeding we introduce some notations and definitions. The degree of a vertex in a graph is denoted by . A vertex of degree is called a -vertex. A -neighbor of is a -vertex adjacent to . is the set of the neighbors of . A -thread of is a path with for . Two edges are at distance at most 2 if either they are adjacent or they are adjacent to a common edge. Denote by the set of edges at distance at most 2 from the edge . Define as the set of colors used by edges in . Denote .

2. Lemmas

Lemma 3. Let be the graph obtained from a path by adding two vertices , so that is adjacent to and is adjacent to . Let be an edge list of . If for , , , and , then has a strong -edge coloring.

Proof. If there is a color , then we first color the edges and with the color . So we can further color the rest of edges with the available colors in the order of , , , and .
Now we assume that . Denote and . If there is a color , we first color the edge with the color . Then after coloring all the other edges, the edge still has one color available. Therefore, we can color the edges with an available color in the order of , , , and . Thus we can further assume . Similarly, we can also assume that . If there is a color , then we first color the edges and with the color . So we can further color the rest of edges with the available colors in the order of , , , and . Therefore we can assume that . We first color the edges , and with the colors , , , and , respectively. Then , , and thus we can further color the edges and . This completes the proof of the lemma.

Lemma 4. Let be a path and an edge list so that if and . Then has a strong -edge coloring.

Proof. We only need to prove the lemma when each is equal to its lower bound. If there is a color , we color both and with . Then and , so we can further color the edges and .
Now assume that . Denote and . If , we first color with and the edge with and then color the edge with an available color. Since , there is still one color available for the edge . Thus has a strong -edge coloring. Similarly we can obtain a strong -edge coloring of if . Now we further assume . That is, , , and are mutually disjoint, and it is easy to see that has a strong -edge coloring.

3. Proof of (i) of Theorem 2

Let be a counterexample with as small as possible. Then there exists a 6-edge list such that is not strongly -edge colorable. We can assume that is connected; otherwise, we can color independently each connected component. A 3-vertex is bad if it is adjacent to a 1-vertex; otherwise it is good.

Claim 1. A 1-vertex is adjacent to a 3-vertex in and each bad 3-vertex is adjacent to two 3-vertices.

Proof. Let be a 1-vertex and its neighbor. Since is a minimum counterexample, has a strong -edge coloring. If or is adjacent to only one 3-vertex, we have , and thus we can easily extend the coloring to , a contradiction.

Claim 2. does not contain a -thread with .

Proof. Suppose that contains a -thread with . Then has a strong -edge coloring by the minimality of . It is easy to see that and . Hence we can extend the coloring of to , a contradiction.

Claim 3. does not contain a path such that are all bad 3-vertices.

Proof. Suppose that contains such a path . Let be the 1-neighbors of , respectively. By the minimality of , has a strong -edge coloring . Since , we have for each , , , and . By Lemma 3, we can further extend the coloring to the rest of the edges of to obtain a strong -edge coloring of , a contradiction.

Claim 4. does not contain the following three configurations:(1)a path such that , , , , , and are bad -vertices, is a good vertex, and another neighbor of is a -vertex (see Figure 1),(2)a path such that , , , , , and are bad -vertices, is a good vertex, and another neighbor of is a bad -vertex (see Figure 2),(3)a path such that , , , , and are bad -vertices, is a good vertex, and another neighbor of is a bad 3-vertex which is also adjacent to a bad 3-vertex (see Figure 3).

316501.fig.001
Figure 1: The configuration of Claim 4(1).
316501.fig.002
Figure 2: The configuration of Claim 4(2).
316501.fig.003
Figure 3: The configuration of Claim 4(3).

Proof. (1) Suppose that there exists a path such that , , , , , and are bad 3-vertices, is a good vertex, and is adjacent to a 2-vertex (see Figure 1). Denote , , , , , and , where , , , , , are 1-vertices, and , , , and . Then has a strong -edge coloring by the minimality of . We are going to extend the coloring to . For each uncolored edge in , we use to denote the set of colors available for . Then has at least three colors because for each edge in . Similarly we have the following:(1) if ,(2) if ,(3) if .
By Lemma 3, we only need to(i)either extend the coloring to so that there are still two colors available for ,(ii)or extend the coloring to so that .
If there is a color , we color the edges and with the color . Then we may further color the edges , , , and . This gives a coloring in (i). Thus . Similarly we can show . Denote .
If , we first color the edge with the color , and then by Lemma 3, we can further extend to the edges , , , , , , and . Since the edge is colored with not in , such extension satisfies (ii). Therefore . Similarly we can show that . Denote . We first color the edges , , and with , , and , respectively. Then has two colors. By Lemma 3, we can first extend to the edges , , , , , and . Since , we can further color the edges , , , and in order.
In each case, we can extend the coloring to a strong -edge coloring of , a contradiction. Therefore, the configuration in Figure 1 does not exist.
Similarly we can also show that the configurations in Figures 2 and 3 do not exist either.

Let be the initial charge of for each vertex . Then . We assign a new charge to each vertex according to the following rules.

R1. For each good 3-vertex , if is a path in which is a maximal -thread, then sends to each for , and if is a path in which is a bad 3-vertex for each , sends to each for .

R2. Each bad 3-vertex sends to its 1-neighbor.

Now we consider the new charge for each vertex .(1) If , then by Claim 1, is adjacent to a bad 3-vertex. Thus .(2) If , then by R1, receives in total from some 3-vertices. Thus .(3) If is a bad 3-vertex, then .(4) Assume that is a good 3-vertex. Denote by and the numbers of bad 3-vertices and 2-vertices receiving charges from , respectively. By Claim 2 and Claim 4, we have and . Hence .If , then . Hence .If , then . Hence .If , then . Hence .If , then . Hence .If , then . Hence .

Therefore we have .

4. Proof of (ii) of Theorem 2

Let be a counterexample with as small as possible. Then there exists a 7-edge list such that is not strongly -edge colorable.

Claim 5. There is no 1-vertex in .

Proof. Suppose to the contrary that contains a 1-vertex , such that is its neighbor. Since is a minimum counterexample, has a strong -edge coloring. Hence , we can easily extend this coloring to , a contradiction.

Claim 6. does not contain a -thread with .

Proof. The proof is similar to that of Claim 2 and thus omitted.

Claim 7. does not contain the following two configurations:(1)A path such that and are 2-vertices, is a 3-vertex, and is a 3-vertex which has two 2-neighbors and one 3-neighbor (see Figure 4).(2)A path where and and both and have three 2-neighbors (see Figure 5).

316501.fig.004
Figure 4: The configuration of Claim 7(1).
316501.fig.005
Figure 5: The configuration of Claim 7(2).

Proof. Suppose that contains a path in Figure 4. Let . Since is a minimal counterexample, has a strong -edge coloring. Then , , and , and we can extend the coloring to , a contradiction.
Suppose that contains the configuration in Figure 5, where , , , , and are 2-vertices and and are 3-vertices. Since is a minimum counterexample, has a strong -edge coloring. Now and . We first color the two edges and to obtain a strong -edge coloring of . It is easy to check that , , , and . By Lemma 4, we can further extend the coloring to , a contradiction.

Let be the initial charge of for each vertex . Then . A -vertex is a 2-vertex with 3-neighbors. We assign a new charge to each vertex according to the following rules.R1. Let be a -vertex and a 3-neighbor of . If is adjacent to three 2-vertices, then sends to ; otherwise sends to .R2. Let be a -vertex and be its 3-neighbor. sends to . (1) If is a -vertex, then by R2, receives from its 3-neighbor. Thus .(2) If is a -vertex with two neighbors and and if one of has three 2-neighbors, then by Claim 7 the other one has at most two 2-neighbors. Hence receives from its neighbors. If neither nor has three 2-neighbors, then receives from its neighbors. Therefore .(3) Assume . By Claim 7, we only consider the following three cases: (a) if is adjacent to three -vertices, then sends out to its neighbors; (b) if is adjacent to at most two -vertices, then it sends out at most to its neighbors; (c) if is adjacent to one -vertex, then it sends out to its neighbors. In each case, we have .

Therefore we have .

5. Proof of (iii) of Theorem 2

Let be a counterexample with as small as possible. Then there exists an 8-edge list such that is not strongly -edge colorable.

Claim 8. There is no 1-vertex in .

Proof. The proof is similar to that of Claim 5 and thus omitted.

Claim 9. There are no two adjacent 2-vertices in .

Proof. Suppose that there are two adjacent 2-vertices and . Let be the other neighbor of . Since is a minimum counterexample, has a strong -edge coloring. Then and . Hence, we can extend the coloring to easily, a contradiction.

Claim 10. A 3-vertex is not adjacent to three 2-vertices in .

Proof. Suppose to the contrary that a 3-vertex is adjacent to three 2-vertices , , and . Since is a minimum counterexample, has a strong -edge coloring. Thus , , and . Therefore, we can extend the coloring to , a contradiction.

Claim 11. does not contain a path where , , and are 2-vertices and and are 3-vertices.

Proof. Suppose to the contrary that contains a path , where , , and are 2-vertices and and are 3-vertices. By the minimality of , has a strong -edge coloring. Uncolor and . It is easy to check that , , , and . By Lemma 4, we can extend the coloring to the path to obtain a strong -edge coloring of , a contradiction.

Let be the initial charge of for each vertex . Then . We assign a new charge to each vertex according to the following rule.R. Let be a 3-vertex and the number of 2-neighbors of . sends to each adjacent 2-vertices if .

Obviously if .

Let be a 2-vertex and its neighbors. By Claims 9 and 10, both and are 3-vertices and have at most two 2-neighbors. If one 3-neighbor of is adjacent to two 2-vertices, then by Claim 11, the other neighbor of is adjacent to only one 2-vertex. Hence receives from its neighbors. If each neighbor of is adjacent to only one 2-vertex, then receives from its neighbors. Hence .

Therefore we have . This contradiction completes the proof.

6. Proof of (iv) of Theorem 2

Let be a counterexample with as small as possible. Then there exists a 9-edge list such that is not strongly -edge colorable.

Claim 12. There is no 1-vertex in .

Proof. The proof is similar to that of Claim 5 and thus omitted.

Claim 13. There are no two adjacent 2-vertices in .

Proof. The proof is similar to that of Claim 9 and thus omitted.

Claim 14. No 3-vertex is adjacent to two 2-vertices in .

Proof. Suppose to the contrary that a 3-vertex is adjacent to two 2-vertices , . Let be the third neighbor of . Since is a minimum counterexample, has a strong -edge coloring. Then , , and . Hence we can extend the coloring to , a contradiction.

Claim 15. If a 3-vertex has a 2-neighbor, then each 3-neighbor of is not adjacent to a 2-vertex.

Proof. Suppose to the contrary that has a 3-neighbor such that is also adjacent to a 2-vertex . Let be the 2-neighbor of . We first assume that . By the choice of , has a strong -edge coloring with the edges and uncolored. Then it is easy to see that and , and thus we may further color the edges and to get a strong -edge coloring of , a contradiction.
Now we assume that . Let be the other neighbor of . Let be a strong -edge coloring of obtained from a strong -edge coloring of by uncoloring the edges and . It is easy to see that , , , and . By Lemma 4, can be extended to those four uncolored edges, and thus has a strong -edge coloring, a contradiction.

Let be the initial charge of for each vertex . Then . We assign a new charge to each vertex according to the following rules.R1. Each -vertex receives from each adjacent vertex.R2. If a 3-vertex is adjacent to a 2-vertex then receives from each 3-neighbor.

Obviously if , then .If and is not adjacent to a 2-vertex, then .If and is adjacent to a 2-vertex, then by Claims 14 and 15, .

Therefore we have . This contradiction completes the proof.

7. Conclusion

This paper studies strong list edge coloring of subcubic graphs. The result can be used to deal with the conflict-free channel assignment problem in wireless radio networks when the admissible channels on the links between transceivers are constrained. We believe that the upper bounds on the maximum average degree in Theorem 2 are not sharp. It would be interesting to find sharp upper bounds for the maximum average degree.

Acknowledgments

The authors acknowledge the support of the National Natural Science Foundation of China (nos. 11101351 and 11171288) and NSF of University in Jiangsu province (no. 11KJB110014).

References

  1. J. L. Fouquet and J. L. Jolivet, “Strong edge-coloring of graphs and applications to multi-k- gons,” Ars Combinatoria A, vol. 16, pp. 141–150, 1983. View at Publisher · View at Google Scholar
  2. J. L. Fouquet and J. L. Jolivet, “Strong edge-coloring of cubic planar graphs,” Progress in Graph Theory, pp. 247–264, 1984.
  3. L. D. Andersen, “The strong chromatic index of a cubic graph is at most 10,” Discrete Mathematics, vol. 108, no. 1–3, pp. 231–252, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  4. K. W. Lih and D. D. F. Liu, “On the strong chromatic index of cubic Halin graphs,” Applied Mathematics Letters, vol. 25, no. 5, pp. 898–901, 2012. View at Publisher · View at Google Scholar
  5. W. C. Shiu and W. K. Tam, “The strong chromatic index of complete cubic Halin graphs,” Applied Mathematics Letters, vol. 22, no. 5, pp. 754–758, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  6. H. Hocquard and P. Valicov, “Strong edge colouring of subcubic graphs,” Discrete Applied Mathematics, vol. 159, no. 15, pp. 1650–1657, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus