Strong List Edge Coloring of Subcubic Graphs

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 𝐺 with vertex set 𝑉(𝐺) and edge set 𝐸(𝐺), a proper edge coloring of 𝐺 is an assignment of colors to the edges of 𝐺 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 𝐺, denoted by 𝜒^′_𝑠(𝐺), is the minimum number of colors needed for a strong edge coloring of 𝐺.

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 Δ the maximum degree of the graph. In 1985, Erdös and Nes̆etr̆il conjectured that the strong chromatic index of a graph is at most (5/4)Δ² when Δ is even and (1/4)(5Δ²−2Δ+1) when Δ is odd. Andersen proved the conjecture for Δ=3 [3]. Strong edge coloring of cubic Halin graphs has been studied in [4, 5].

Let mad(𝐺)=max_{𝐻⊆𝐺,|𝑉(𝐻)|≥1}(2|𝐸(𝐻)|/|𝑉(𝐻)|) be the maximum average degree of the graph 𝐺. Hocquard and Valicov [6] considered the subcubic graphs with bounded maximum average degree, and they proved the following results.

Theorem 1. Let 𝐺 be a subcubic graph.(i)If mad(𝐺)<15/7, then 𝜒^′_𝑠(𝐺)≤6.(ii)If mad(𝐺)<27/11, then 𝜒^′_𝑠(𝐺)≤7.(iii)If mad(𝐺)<13/5, then 𝜒^′_𝑠(𝐺)≤8.(iv)If mad(𝐺)<36/13, then 𝜒^′_𝑠(𝐺)≤9.

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 𝐿 of a graph 𝐺 is a mapping that assigns a finite set to each edge of 𝐺. Denote 𝐿={𝐿(𝑒)∶𝑒∈𝐸(𝐺)}. We say that 𝐿 is a 𝑘-edge list if |𝐿(𝑒)|≥𝑘 for each edge 𝑒 in 𝐺. The graph 𝐺 is strongly 𝐿-edge colorable if there exists a strong edge coloring 𝑐 of 𝐺 such that 𝑐(𝑒)∈𝐿(𝑒) for every edge 𝑒 of 𝐺. For a positive integer 𝑘, a graph 𝐺 is strongly 𝑘-edge choosable if for every 𝑘-edge list 𝐿, 𝐺 is strongly 𝐿-edge colorable. The strong list chromatic index 𝜒^′_𝑙𝑠(𝐺) is the minimum positive integer 𝑘 for which 𝐺 is strongly 𝑘-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 𝐺 be a subcubic graph.(i)If mad(𝐺)<15/7, then 𝜒^′_𝑙𝑠(𝐺)≤6.(ii)If mad(𝐺)<27/11, then 𝜒^′_𝑙𝑠(𝐺)≤7.(iii)If mad(𝐺)<13/5, then 𝜒^′_𝑙𝑠(𝐺)≤8.(iv)If mad(𝐺)<36/13, then 𝜒^′_𝑙𝑠(𝐺)≤9.

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 𝑣 in a graph is denoted by 𝑑(𝑣). A vertex of degree 𝑘 is called a 𝑘-vertex. A 𝑘-neighbor of 𝑣 is a 𝑘-vertex adjacent to 𝑣. 𝑁(𝑣) is the set of the neighbors of 𝑣. A 𝑡-thread of 𝐺 is a path 𝑃_𝑡=𝑥₁𝑥₂⋯𝑥_𝑡 with 𝑑(𝑥_𝑖)=2 for 𝑖=1,2,…,𝑡. Two edges are at distance at most 2 if either they are adjacent or they are adjacent to a common edge. Denote by 𝑁₂(𝑒) the set of edges at distance at most 2 from the edge 𝑒. Define LSC(𝑁₂(𝑒)) as the set of colors used by edges in 𝑁₂(𝑒). Denote 𝐿^′(𝑒)=𝐿(𝑒)⧵LSC(𝑁₂(𝑒)).

2. Lemmas

Lemma 3. Let 𝐺 be the graph obtained from a path 𝑥₁𝑥₂𝑥₃𝑥₄𝑥₅ by adding two vertices 𝑢, 𝑣 so that 𝑢 is adjacent to 𝑥₃ and 𝑣 is adjacent to 𝑥₄. Let 𝐿 be an edge list of 𝐺. If |𝐿(𝑥_𝑖𝑥_𝑖+1)|≥3 for 𝑖=1,2,3, |𝐿(𝑥₄𝑥₅)|≥2, |𝐿(𝑥₃𝑢)|≥5, and |𝐿(𝑥₄𝑣)|≥4, then 𝐺 has a strong 𝐿-edge coloring.

Proof. If there is a color 𝑎∈𝐿(𝑥₁𝑥₂)∩𝐿(𝑥₄𝑥₅), then we first color the edges 𝑥₁𝑥₂ and 𝑥₄𝑥₅ with the color 𝑎. So we can further color the rest of edges with the available colors in the order of 𝑥₂𝑥₃, 𝑥₃𝑥₄, 𝑥₄𝑣, and 𝑥₃𝑢.
Now we assume that 𝐿(𝑥₁𝑥₂)∩𝐿(𝑥₄𝑥₅)=∅. Denote 𝐿(𝑥₁𝑥₂)={𝑎,𝑏,𝑐} and 𝐿(𝑥₄𝑥₅)={𝑑,𝑒}. If there is a color 𝑡∈𝐿(𝑥₂𝑥₃)⧵{𝑎,𝑏,𝑐}, we first color the edge 𝑥₂𝑥₃ with the color 𝑡. Then after coloring all the other edges, the edge 𝑥₁𝑥₂ still has one color available. Therefore, we can color the edges with an available color in the order of 𝑥₄𝑥₅, 𝑥₃𝑥₄, 𝑥₄𝑣, and 𝑥₃𝑢. Thus we can further assume 𝐿(𝑥₂𝑥₃)={𝑎,𝑏,𝑐}. Similarly, we can also assume that 𝐿(𝑥₃𝑥₄)={𝑎,𝑏,𝑐}. If there is a color 𝑠∈{𝑎,𝑏,𝑐}∩𝐿(𝑥₄𝑣), then we first color the edges 𝑥₁𝑥₂ and 𝑥₄𝑣 with the color 𝑠. So we can further color the rest of edges with the available colors in the order of 𝑥₂𝑥₃, 𝑥₃𝑥₄, 𝑥₃𝑢, and 𝑥₄𝑥₅. Therefore we can assume that {𝑎,𝑏,𝑐}∩𝐿(𝑥₄𝑣)=∅. We first color the edges 𝑥₁𝑥₂,𝑥₂𝑥₃,𝑥₃𝑥₄, and 𝑥₄𝑥₅ with the colors 𝑎, 𝑏, 𝑐, and 𝑑, respectively. Then |𝐿^′(𝑥₃𝑢)|≥1, |𝐿^′(𝑥₄𝑣)|≥3, and thus we can further color the edges 𝑥₃𝑢 and 𝑥₄𝑣. This completes the proof of the lemma.

Lemma 4. Let 𝑃=𝑥𝑦𝑧𝑢𝑣 be a path and 𝐿 an edge list so that |𝐿(𝑒)|≥2 if 𝑒∈{𝑥𝑦,𝑧𝑢,𝑢𝑣} and |𝐿(𝑦𝑧)|≥3. Then 𝑃 has a strong 𝐿-edge coloring.

Proof. We only need to prove the lemma when each |𝐿(𝑒)| is equal to its lower bound. If there is a color 𝑎∈𝐿(𝑥𝑦)∩𝐿(𝑢𝑣), we color both 𝑥𝑦 and 𝑢𝑣 with 𝑎. Then |𝐿^′(𝑧𝑢)|≥1 and |𝐿^′(𝑦𝑧)|≥2, so we can further color the edges 𝑧𝑢 and 𝑦𝑧.
Now assume that 𝐿(𝑥𝑦)∩𝐿(𝑢𝑣)=∅. Denote 𝐿(𝑥𝑦)={𝑎,𝑏} and 𝐿(𝑢𝑣)={𝑐,𝑑}. If 𝑎∈𝐿(𝑧𝑢), we first color 𝑧𝑢 with 𝑎 and the edge 𝑥𝑦 with 𝑏 and then color the edge 𝑦𝑧 with an available color. Since 𝑎,𝑏∉𝐿(𝑢𝑣), there is still one color available for the edge 𝑢𝑣. Thus 𝑃 has a strong 𝐿-edge coloring. Similarly we can obtain a strong 𝐿-edge coloring of 𝑃 if 𝐿(𝑧𝑢)∩𝐿(𝑢𝑣)≠∅. Now we further assume 𝐿(𝑧𝑢)∩[𝐿(𝑥𝑦)∪𝐿(𝑢𝑣)]=∅. That is, 𝐿(𝑥𝑦), 𝐿(𝑧𝑢), and 𝐿(𝑢𝑣) are mutually disjoint, and it is easy to see that 𝑃 has a strong 𝐿-edge coloring.

3. Proof of (i) of Theorem 2

Let 𝐻 be a counterexample with |𝐸(𝐻)| as small as possible. Then there exists a 6-edge list 𝐿 such that 𝐻 is not strongly 𝐿-edge colorable. We can assume that 𝐻 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 𝐻 and each bad 3-vertex is adjacent to two 3-vertices.

Proof. Let 𝑢 be a 1-vertex and 𝑣 its neighbor. Since 𝐻 is a minimum counterexample, 𝐻⧵{𝑢𝑣} has a strong 𝐿-edge coloring. If 𝑑(𝑣)=2 or 𝑣 is adjacent to only one 3-vertex, we have |𝐿^′(𝑢𝑣)|≥1, and thus we can easily extend the coloring to 𝐻, a contradiction.

Claim 2. 𝐻 does not contain a 𝑡-thread with 𝑡≥3.

Proof. Suppose that 𝐻 contains a 𝑡-thread 𝑥₁𝑥₂⋯𝑥_𝑡 with 𝑡≥3. Then 𝐻^′=𝐻⧵{𝑥₁𝑥₂,𝑥₂𝑥₃} has a strong 𝐿-edge coloring by the minimality of 𝐻. It is easy to see that |𝐿^′(𝑥₁𝑥₂)|≥2 and |𝐿^′(𝑥₂𝑥₃)|≥2. Hence we can extend the coloring of 𝐻^′ to 𝐻, a contradiction.

Claim 3. 𝐻 does not contain a path 𝑥₂𝑥₃𝑥₄𝑥₅ such that 𝑥₂,𝑥₃,𝑥₄,𝑥₅ are all bad 3-vertices.

Proof. Suppose that 𝐻 contains such a path 𝑥₂𝑥₃𝑥₄𝑥₅. Let 𝑥₁,𝑢,𝑣 be the 1-neighbors of 𝑥₂,𝑥₃,𝑥₄, respectively. By the minimality of 𝐻, 𝐻⧵[{𝑥_𝑖𝑥_𝑖+1∣𝑖=1,…,4}∪{𝑥₃𝑢,𝑥₄𝑣}] has a strong 𝐿-edge coloring 𝑓. Since |𝐿(𝑒)|≥6, we have |𝐿^′(𝑥_𝑖𝑥_𝑖+1)|≥3 for each 𝑖=1,2,3, |𝐿^′(𝑥₄𝑥₅)|≥2, |𝐿^′(𝑥₃𝑢)|≥5, and |𝐿^′(𝑥₄𝑣)|≥4. By Lemma 3, we can further extend the coloring to the rest of the edges of 𝐻 to obtain a strong 𝐿-edge coloring of 𝐻, a contradiction.

Claim 4. 𝐻 does not contain the following three configurations:(1)a path 𝑥𝑢𝑣𝑤𝑦𝑟𝑠𝑡𝑡₂ such that 𝑢, 𝑣, 𝑤, 𝑟, 𝑠, and 𝑡 are bad 3-vertices, 𝑦 is a good vertex, and another neighbor 𝑧 of 𝑦 is a 2-vertex (see Figure 1),(2)a path 𝑥𝑢𝑣𝑤𝑦𝑟𝑠𝑡𝑡₂ such that 𝑢, 𝑣, 𝑤, 𝑟, 𝑠, and 𝑡 are bad 3-vertices, 𝑦 is a good vertex, and another neighbor 𝑧 of 𝑦 is a bad 3-vertex (see Figure 2),(3)a path 𝑥𝑢𝑣𝑤𝑦𝑟𝑠 such that 𝑢, 𝑣, 𝑤, 𝑟, and 𝑠 are bad 3-vertices, 𝑦 is a good vertex, and another neighbor 𝑧 of 𝑦 is a bad 3-vertex which is also adjacent to a bad 3-vertex (see Figure 3).

Figure 1 

The configuration of Claim 4(1).

Figure 2 

The configuration of Claim 4(2).

Figure 3 

The configuration of Claim 4(3).

Proof. (1) Suppose that there exists a path 𝑥𝑢𝑣𝑤𝑦𝑟𝑠𝑡𝑡₂ such that 𝑢, 𝑣, 𝑤, 𝑟, 𝑠, and 𝑡 are bad 3-vertices, 𝑦 is a good vertex, and 𝑦 is adjacent to a 2-vertex (see Figure 1). Denote 𝑁(𝑢)={𝑥,𝑢₁,𝑣}, 𝑁(𝑣)={𝑢,𝑣₁,𝑤}, 𝑁(𝑤)={𝑣,𝑤₁,𝑦}, 𝑁(𝑟)={𝑦,𝑟₁,𝑠}, 𝑁(𝑠)={𝑟,𝑠₁,𝑡}, and 𝑁(𝑡)={𝑠,𝑡₁,𝑡₂}, where 𝑢₁, 𝑣₁, 𝑤₁, 𝑟₁, 𝑠₁, 𝑡₁ are 1-vertices, and 𝑁(𝑥)={𝑢,𝑥₁,𝑥₂}, 𝑁(𝑦)={𝑤,𝑟,𝑧}, 𝑁(𝑧)={𝑦,𝑧₁}, and 𝑁(𝑡₂)={𝑡,𝑡₃,𝑡₄}. Then 𝐻⧵{𝑢𝑢₁,𝑢𝑣,𝑣𝑣₁,𝑣𝑤,𝑤𝑤₁,𝑤𝑦,𝑦𝑧,𝑦𝑟,𝑟𝑟₁,𝑟𝑠,𝑠𝑠₁,𝑠𝑡,𝑡𝑡₁} has a strong 𝐿-edge coloring 𝐶 by the minimality of 𝐻. We are going to extend the coloring 𝐶 to 𝐻. For each uncolored edge 𝑒 in 𝐻, we use 𝐿^′(𝑒) to denote the set of colors available for 𝑒. Then 𝐿^′(𝑢𝑢₁)=𝐿(𝑢𝑢₁)⧵{𝐶(𝑥𝑢),𝐶(𝑥𝑥₁),𝐶(𝑥𝑥₂)} has at least three colors because |𝐿(𝑒)|≥6 for each edge 𝑒 in 𝐻. Similarly we have the following:(1)|𝐿^′(𝑒)|≥3 if 𝑒∈{𝑢𝑢₁,𝑢𝑣,𝑠𝑡,𝑡𝑡₁,𝑦𝑧},(2)|𝐿^′(𝑒)|≥5 if 𝑒∈{𝑣𝑣₁,𝑣𝑤,𝑤𝑦,𝑦𝑟,𝑟𝑠,𝑠𝑠₁},(3)|𝐿^′(𝑒)≥6 if 𝑒∈{𝑤𝑤₁,𝑟𝑟₁}.
By Lemma 3, we only need to(i)either extend the coloring 𝐶 to 𝑦𝑧,𝑦𝑟,𝑟𝑠,𝑠𝑠₁,𝑠𝑡,𝑡𝑡₁ so that there are still two colors available for 𝑟𝑟₁,(ii)or extend the coloring 𝐶 to 𝑦𝑧,𝑦𝑟,𝑟𝑟₁,𝑟𝑠,𝑠𝑠₁,𝑠𝑡,𝑡𝑡₁ so that |{𝐶(𝑦𝑧),𝐶(𝑦𝑟),𝐶(𝑟𝑟₁),𝐶(𝑟𝑠)}∩[𝐿(𝑤𝑦)⧵{𝐶(𝑧𝑧₁)}]|≤3.
If there is a color 𝑎∈𝐿^′(𝑦𝑧)∩𝐿^′(𝑠𝑡), we color the edges 𝑦𝑧 and 𝑠𝑡 with the color 𝑎. Then we may further color the edges 𝑡𝑡₁, 𝑠𝑠₁, 𝑟𝑠, and 𝑦𝑟. This gives a coloring in (i). Thus 𝐿^′(𝑦𝑧)∩𝐿^′(𝑠𝑡)=∅. Similarly we can show 𝐿^′(𝑦𝑧)∩𝐿^′(𝑠𝑠₁)=∅. Denote 𝐿^′(𝑦𝑧)={𝑎,𝑏,𝑐}.
If 𝑎∉𝐿^′(𝑦𝑤), we first color the edge 𝑦𝑧 with the color 𝑎, and then by Lemma 3, we can further extend 𝐶 to the edges 𝑦𝑧, 𝑦𝑟, 𝑟𝑟₁, 𝑟𝑠, 𝑠𝑠₁, 𝑠𝑡, and 𝑡𝑡₁. Since the edge 𝑦𝑧 is colored with 𝑎 not in 𝐿^′(𝑤𝑦), such extension satisfies (ii). Therefore 𝐿^′(𝑦𝑧)⊆𝐿^′(𝑤𝑦). Similarly we can show that 𝐿^′(𝑤𝑦)=𝐿^′(𝑦𝑟)=𝐿^′(𝑟𝑠). Denote 𝐿^′(𝑤𝑦)={𝑎,𝑏,𝑐,𝑑,𝑒}. We first color the edges 𝑦𝑧, 𝑦𝑟, and 𝑟𝑠 with 𝑎, 𝑏, and 𝑐, respectively. Then 𝐿^′(𝑤𝑦)⧵{𝑎,𝑏,𝑐} has two colors. By Lemma 3, we can first extend 𝐶 to the edges 𝑢𝑢₁, 𝑢𝑣, 𝑣𝑣₁, 𝑣𝑤, 𝑤𝑤₁, and 𝑤𝑦. Since {𝑎,𝑏,𝑐}∩[𝐿^′(𝑠𝑡)∪𝐿^′(𝑠𝑠₁)]=∅, we can further color the edges 𝑡𝑡₁, 𝑠𝑡, 𝑟𝑟₁, and 𝑠𝑠₁ in order.
In each case, we can extend the coloring 𝐶 to a strong 𝐿-edge coloring of 𝐻, 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 𝑀(𝑥)=𝑑(𝑥)−(15/7) be the initial charge of 𝑥 for each vertex 𝑥. Then ∑_{𝑥∈𝑉(𝐻)}𝑀(𝑥)<0. We assign a new charge to each vertex according to the following rules.

R1. For each good 3-vertex 𝑥, if 𝑥𝑥₁𝑥₂⋯𝑥_𝑡𝑥_𝑡+1 is a path in which 𝑥₁𝑥₂⋯𝑥_𝑡 is a maximal 𝑡-thread, then 𝑥 sends 1/14 to each 𝑥_𝑖 for 𝑖=1,…,𝑡, and if 𝑥𝑥₁𝑥₂⋯𝑥_𝑡𝑥_𝑡+1 is a path in which 𝑥_𝑖 is a bad 3-vertex for each 𝑖=1,…,𝑡, 𝑥 sends 1/7 to each 𝑥_𝑖 for 𝑖=1,…,𝑡.

R2. Each bad 3-vertex sends 8/7 to its 1-neighbor.

Now we consider the new charge 𝑀^′(𝑥) for each vertex 𝑥.(1) If 𝑑(𝑥)=1, then by Claim 1, 𝑥 is adjacent to a bad 3-vertex. Thus 𝑀^′(𝑥)=1−(15/7)+(8/7)=0.(2) If 𝑑(𝑥)=2, then by R1, 𝑥 receives 2×(1/14)=(1/7) in total from some 3-vertices. Thus 𝑀^′(𝑥)=2−(15/7)+(1/7)=0.(3) If 𝑥 is a bad 3-vertex, then 𝑀^′(𝑥)=(8/7)−(8/7)=0.(4) Assume that 𝑥 is a good 3-vertex. Denote by 𝑡 and 𝑠 the numbers of bad 3-vertices and 2-vertices receiving charges from 𝑥, respectively. By Claim 2 and Claim 4, we have 0≤𝑡≤6 and 0≤𝑠≤6. Hence 𝑀^′(𝑥)=3−(15/7)−(𝑡/7)−(𝑠/14)=(6/7)−(𝑡/7)−(𝑠/14). If 𝑡=0, then 𝑠≤6. Hence 𝑀^′(𝑥)≥(6/7)−(6/14)>0. If 0<𝑡≤3, then 𝑠≤4. Hence 𝑀^′(𝑥)≥(6/7)−(3/7)−(4/14)>0. If 𝑡=4, then 𝑠≤2. Hence 𝑀^′(𝑥)≥(6/7)−(4/7)−(2/14)>0. If 𝑡=5, then 𝑠≤2. Hence 𝑀^′(𝑥)≥(6/7)−(5/7)−(2/14)≥0. If 𝑡=6, then 𝑠=0. Hence 𝑀^′(𝑥)≥(6/7)−(6/7)≥0.

Therefore we have 0≤∑_{𝑥∈𝑉(𝐻)}𝑀^′(𝑥)=∑_{𝑥∈𝑉(𝐻)}𝑀(𝑥)<0.

4. Proof of (ii) of Theorem 2

Let 𝐻 be a counterexample with |𝐸(𝐻)| as small as possible. Then there exists a 7-edge list 𝐿 such that 𝐻 is not strongly 𝐿-edge colorable.

Claim 5. There is no 1-vertex in 𝐻.

Proof. Suppose to the contrary that 𝐻 contains a 1-vertex 𝑢, such that 𝑣 is its neighbor. Since 𝐻 is a minimum counterexample, 𝐻^′=𝐻⧵{𝑢𝑣} has a strong 𝐿-edge coloring. Hence |𝐿^′(𝑢𝑣)|≥1, we can easily extend this coloring to 𝐻, a contradiction.

Claim 6. 𝐻 does not contain a 𝑡-thread with 𝑡≥3.

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

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

Figure 4 

The configuration of Claim 7(1).

Figure 5 

The configuration of Claim 7(2).

Proof. (1) Suppose that 𝐻 contains a path 𝑥𝑢𝑣𝑤 in Figure 4. Let 𝐻^′=𝐻⧵{𝑢𝑣,𝑣𝑤,𝑤𝑦}. Since 𝐻 is a minimal counterexample, 𝐻^′ has a strong 𝐿-edge coloring. Then |𝐿^′(𝑤𝑦)|≥1, |𝐿^′(𝑣𝑤)|≥2, and |𝐿^′(𝑢𝑣)|≥3, and we can extend the coloring to 𝐻, a contradiction.
(2) Suppose that 𝐻 contains the configuration in Figure 5, where 𝑦, 𝑢₁, 𝑤, 𝑣₁, and 𝑧 are 2-vertices and 𝑢 and 𝑣 are 3-vertices. Since 𝐻 is a minimum counterexample, 𝐻^′=𝐻⧵{𝑦𝑢,𝑢𝑢₁,𝑢𝑤,𝑤𝑣,𝑣𝑣₁,𝑣𝑧} has a strong 𝐿-edge coloring. Now |𝐿^′(𝑢𝑢₁)|≥3 and |𝐿^′(𝑣𝑣₁)|≥3. We first color the two edges 𝑢𝑢₁ and 𝑣𝑣₁ to obtain a strong 𝐿-edge coloring of 𝐻⧵{𝑦𝑢,𝑢𝑤,𝑤𝑣,𝑣𝑧}. It is easy to check that |𝐿^′(𝑦𝑢)|≥2, |𝐿^′(𝑢𝑤)|≥3, |𝐿^′(𝑤𝑣)|≥3, and |𝐿^′(𝑣𝑧)|≥2. By Lemma 4, we can further extend the coloring to 𝐻, a contradiction.

Let 𝑀(𝑥)=𝑑(𝑥)−(27/11) be the initial charge of 𝑥 for each vertex 𝑥. Then ∑_{𝑥∈𝑉(𝐻)}𝑀(𝑥)<0. A 2_𝑘-vertex is a 2-vertex with 𝑘 3-neighbors. We assign a new charge to each vertex according to the following rules. R1. Let 𝑥 be a 2₂-vertex and 𝑢 a 3-neighbor of 𝑥. If 𝑢 is adjacent to three 2-vertices, then 𝑢 sends 2/11 to 𝑥; otherwise 𝑢 sends 3/11 to 𝑥. R2. Let 𝑥 be a 2₁-vertex and 𝑢 be its 3-neighbor. 𝑢 sends 5/11 to 𝑥. (1) If 𝑥 is a 2₁-vertex, then by R2, 𝑥 receives 5/11 from its 3-neighbor. Thus 𝑀^′(𝑥)=2−(27/11)+(5/11)=0.(2) If 𝑥 is a 2₂-vertex with two neighbors 𝑢 and 𝑣 and if one of 𝑢,𝑣 has three 2-neighbors, then by Claim 7 the other one has at most two 2-neighbors. Hence 𝑥 receives (2/11)+(3/11)=5/11 from its neighbors. If neither 𝑢 nor 𝑣 has three 2-neighbors, then 𝑥 receives (3/11)+(3/11)=6/11 from its neighbors. Therefore 𝑀^′(𝑥)≥2−(27/11)+(5/11)≥0.(3) Assume 𝑑(𝑥)=3. By Claim 7, we only consider the following three cases: (a) if 𝑥 is adjacent to three 2₂-vertices, then 𝑥 sends out 3×(2/11)=6/11 to its neighbors; (b) if 𝑥 is adjacent to at most two 2₂-vertices, then it sends out at most 2×(3/11)=6/11 to its neighbors; (c) if 𝑥 is adjacent to one 2₁-vertex, then it sends out 5/11 to its neighbors. In each case, we have 𝑀^′(𝑥)≥3−(27/11)−(6/11)=0.

Therefore we have 0≤∑_{𝑥∈𝑉(𝐻)}𝑀^′(𝑥)=∑_{𝑥∈𝑉(𝐻)}𝑀(𝑥)<0.

5. Proof of (iii) of Theorem 2

Let 𝐻 be a counterexample with |𝐸(𝐻)| as small as possible. Then there exists an 8-edge list 𝐿 such that 𝐻 is not strongly 𝐿-edge colorable.

Claim 8. There is no 1-vertex in 𝐻.

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

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

Proof. Suppose that there are two adjacent 2-vertices 𝑢 and 𝑣. Let 𝑤 be the other neighbor of 𝑣. Since 𝐻 is a minimum counterexample, 𝐻⧵{𝑢𝑣,𝑣𝑤} has a strong 𝐿-edge coloring. Then |𝐿^′(𝑢𝑣)|≥3 and |𝐿^′(𝑣𝑤)|≥1. Hence, we can extend the coloring to 𝐻 easily, a contradiction.

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

Proof. Suppose to the contrary that a 3-vertex 𝑢 is adjacent to three 2-vertices 𝑥, 𝑣, and 𝑤. Since 𝐻 is a minimum counterexample, 𝐻⧵{𝑥𝑢,𝑢𝑣,𝑢𝑤} has a strong 𝐿-edge coloring. Thus |𝐿^′(𝑥𝑢)|≥3, |𝐿^′(𝑢𝑣)|≥3, and |𝐿^′(𝑢𝑤)|≥3. Therefore, we can extend the coloring to 𝐻, a contradiction.

Claim 11. 𝐻 does not contain a path 𝑃=𝑢𝑣𝑤𝑥𝑦 where 𝑢, 𝑤, and 𝑦 are 2-vertices and 𝑣 and 𝑥 are 3-vertices.

Proof. Suppose to the contrary that 𝐻 contains a path 𝑢𝑣𝑤𝑥𝑦, where 𝑢, 𝑤, and 𝑦 are 2-vertices and 𝑣 and 𝑥 are 3-vertices. By the minimality of 𝐻, 𝐻⧵{𝑣𝑤,𝑤𝑥} has a strong 𝐿-edge coloring. Uncolor 𝑢𝑣 and 𝑥𝑦. It is easy to check that |𝐿^′(𝑢𝑣)|≥2, |𝐿^′(𝑥𝑦)|≥2, |𝐿^′(𝑣𝑤)|≥3, and |𝐿^′(𝑤𝑥)|≥3. By Lemma 4, we can extend the coloring to the path 𝑢𝑣𝑤𝑥𝑦 to obtain a strong 𝐿-edge coloring of 𝐻, a contradiction.

Let 𝑀(𝑥)=𝑑(𝑥)−(13/5) be the initial charge of 𝑥 for each vertex 𝑥. Then ∑_{𝑥∈𝑉(𝐻)}𝑀(𝑥)<0. We assign a new charge to each vertex according to the following rule. R. Let 𝑥 be a 3-vertex and 𝑡 the number of 2-neighbors of 𝑥. 𝑥 sends 2/5𝑡 to each adjacent 2-vertices if 𝑡≠0.

Obviously 𝑀^′(𝑥)≥0 if 𝑑(𝑥)=3.

Let 𝑥 be a 2-vertex and 𝑢,𝑣 its neighbors. By Claims 9 and 10, both 𝑢 and 𝑣 are 3-vertices and have at most two 2-neighbors. If one 3-neighbor of 𝑥 is adjacent to two 2-vertices, then by Claim 11, the other neighbor of 𝑥 is adjacent to only one 2-vertex. Hence 𝑥 receives (1/5)+(2/5)=3/5 from its neighbors. If each neighbor of 𝑥 is adjacent to only one 2-vertex, then 𝑥 receives (2/5)+(2/5)=4/5 from its neighbors. Hence 𝑀^′(𝑥)≥2−(13/5)+(3/5)=0.

Therefore we have 0≤∑_{𝑥∈𝑉(𝐻)}𝑀^′(𝑥)=∑_{𝑥∈𝑉(𝐻)}𝑀(𝑥)<0. This contradiction completes the proof.

6. Proof of (iv) of Theorem 2

Let 𝐻 be a counterexample with |𝐸(𝐻)| as small as possible. Then there exists a 9-edge list 𝐿 such that 𝐻 is not strongly 𝐿-edge colorable.

Claim 12. There is no 1-vertex in 𝐻.

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

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

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 𝐻.

Proof. Suppose to the contrary that a 3-vertex 𝑢 is adjacent to two 2-vertices 𝑣, 𝑤. Let 𝑥 be the third neighbor of 𝑢. Since 𝐻 is a minimum counterexample, 𝐻⧵{𝑢𝑣,𝑢𝑤,𝑢𝑥} has a strong 𝐿-edge coloring. Then |𝐿^′(𝑢𝑥)|≥1, |𝐿^′(𝑢𝑣)|≥3, and |𝐿^′(𝑢𝑤)|≥3. Hence we can extend the coloring to 𝐻, a contradiction.

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

Proof. Suppose to the contrary that 𝑥 has a 3-neighbor 𝑦 such that 𝑦 is also adjacent to a 2-vertex 𝑢. Let 𝑧 be the 2-neighbor of 𝑥. We first assume that 𝑢=𝑧. By the choice of 𝐻, 𝐻⧵{𝑥𝑧,𝑦𝑧} has a strong 𝐿-edge coloring with the edges 𝑥𝑧 and 𝑦𝑧 uncolored. Then it is easy to see that |𝐿^′(𝑥𝑧)|≥3 and |𝐿^′(𝑦𝑧)|≥3, and thus we may further color the edges 𝑥𝑧 and 𝑦𝑧 to get a strong 𝐿-edge coloring of 𝐻, a contradiction.
Now we assume that 𝑢≠𝑧. Let 𝑣 be the other neighbor of 𝑧. Let 𝐶 be a strong 𝐿-edge coloring of 𝐻⧵{𝑣𝑧,𝑧𝑥,𝑥𝑦,𝑦𝑢} obtained from a strong 𝐿-edge coloring of 𝐻−{𝑧} by uncoloring the edges 𝑥𝑦 and 𝑦𝑢. It is easy to see that |𝐿^′(𝑣𝑧)|≥2, |𝐿^′(𝑧𝑥)|≥3, |𝐿^′(𝑥𝑦)|≥2, and |𝐿^′(𝑦𝑢)|≥2. By Lemma 4, 𝐶 can be extended to those four uncolored edges, and thus 𝐻 has a strong 𝐿-edge coloring, a contradiction.

Let 𝑀(𝑥)=𝑑(𝑥)−(36/13) be the initial charge of 𝑥 for each vertex 𝑥. Then ∑_{𝑥∈𝑉(𝐻)}𝑀(𝑥)<0. We assign a new charge to each vertex according to the following rules. R1. Each 2-vertex receives 5/13 from each adjacent vertex. R2. If a 3-vertex 𝑥 is adjacent to a 2-vertex then 𝑥 receives 1/13 from each 3-neighbor.

Obviously if 𝑑(𝑥)=2, then 𝑀^′(𝑥)=2−(36/13)+(10/13)=0. If 𝑑(𝑥)=3 and 𝑥 is not adjacent to a 2-vertex, then 𝑀^′(𝑥)≥3−(36/13)−3×(1/13)=0. If 𝑑(𝑥)=3 and 𝑥 is adjacent to a 2-vertex, then by Claims 14 and 15, 𝑀^′(𝑥)=3−(36/13)−(5/13)+2×(1/13)=0.

Therefore we have 0≤∑_{𝑥∈𝑉(𝐻)}𝑀^′(𝑥)=∑_{𝑥∈𝑉(𝐻)}𝑀(𝑥)<0. 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).