Some Conclusion on Unique 𝑘-List Colorable Complete Multipartite Graphs

Abstract

If a graph 𝐺 admits a 𝑘-list assignment 𝐿 such that 𝐺 has a unique 𝐿-coloring, then 𝐺 is called uniquely 𝑘-list colorable graph, or U𝑘LC graph for short. In the process of characterizing U𝑘LC graphs, the complete multipartite graphs 𝐾_{1∗𝑟,𝑠}(𝑟,𝑠∈𝑁) are often researched. But it is usually not easy to construct the unique 𝑘-list assignment of 𝐾_{1∗𝑟,𝑠}. In this paper, we give some propositions about the property of the graph 𝐾_{1∗𝑟,𝑠} when it is U𝑘LC, which provide a very significant guide for constructing such list assignment. Then a special example of U𝑘LC graphs 𝐾_{1∗𝑟,𝑠} as a application of these propositions is introduced. The conclusion will pave the way to characterize U𝑘LC complete multipartite graphs.

1. Introduction

In this section, some definitions and results about list colorings which are referred to throughout the paper are introduced. For the necessary definitions and notation, we refer the reader to standard texts, such as [1]. Following the paper [2], we use the notation 𝐾_𝑠∗𝑟(𝑟,𝑠∈𝑁, 𝑁 is the set of natural numbers) for a complete 𝑟-partite graph in which each part is of size 𝑠. Notation such as 𝐾_{𝑠∗𝑟,𝑡},(𝑟,𝑠,𝑡∈𝑁) is used similarly.

The idea of list colorings of graphs is due, independently, to Vizing [3] and Erdős et al. [4]. For a graph 𝐺=(𝑉,𝐸) and each vertex 𝑣∈𝑉(𝐺), let 𝐿(𝑣) denote a list of colors available for 𝑣. 𝐿={𝐿(𝑣)∣𝑣∈𝑉(𝐺)} is said to be a list assignment of 𝐺. If |𝐿(𝑣)|=𝑘 for all 𝑣∈𝑉(𝐺), then 𝐿 is called 𝑘-list assignment of 𝐺. For example, the numbers nearby the vertices in Figure 1 are 2-list assignment of the graph. A list coloring from a given collection of lists is a proper coloring 𝑐 such that 𝑐(𝑣) is chosen from 𝐿(𝑣). We will refer to such a coloring as an L-coloring [5]. In Figure 1, the set of circled numbers makes a 2-list coloring of the graph.

Figure 1 

The list of graph.

The list coloring model can be used in the channel assignment [6–8]. The fixed channel allocation scheme leads to low channel utilization across the whole channel. It requires a more effective channel assignment and management policy, which allows unused parts of channel to become available temporarily for other usages so that the scarcity of the channel can be largely mitigated [6]. It is a discrete optimization problem. A model for channel availability observed by the secondary users is introduced in [6]. We abstract each secondary network topology into a graph, where vertices represent wireless users such as wireless lines, WLANs, or cells, and edges represent interferences between vertices. In particular, if two vertices are connected by an edge in the graph, we assume that these two vertices cannot use the same spectrum simultaneously. In addition, we associate with each vertex a set, which represents the available spectra at this location. Due to the differences in the geographical location of each vertex, the sets of spectra of different nodes may be different. Then a list coloring model is constructed.

The research of list coloring consists of two parts: the choosability and the unique list colorability. Some relations between uniquely list colorability and choosability of a graph are presented in [9]. In this paper, we research the unique list colorability of graph.

The concept of unique list coloring was introduced by Dinitz and Martin [10] and independently by Mahmoodian and Mahdian [11], which can be used to study defining set of 𝑘-coloring [12] and critical sets in Latin squares [13]. Let 𝐺 be a graph with 𝑛 vertices, and suppose that for each vertex 𝑣 in 𝐺, there exists a list of 𝑘 colors 𝐿(𝑣), such that there exists a unique 𝐿-coloring for 𝐺; then 𝐺 is called uniquely k-list colorable graph or a U𝑘LC graph for short. It is obvious that the set of circled numbers makes a 2-list coloring of the graph in Figure 2. For a graph 𝐺, it is said to have the property 𝑀(𝑘) if and only if it is not uniquely 𝑘-list colorable graph. So 𝐺 has the property 𝑀(𝑘) if for any collection of lists assigned to its vertices, each of size 𝑘, either there is no list coloring for 𝐺 or there exist two list colorings. Note that the 𝑚-number of a graph 𝐺, denoted by 𝑚(𝐺), is defined to be the least integer 𝑘 such that 𝐺 has the property 𝑀(𝑘).

Figure 2 

The list coloring of graph.

It is clear from the definition of uniquely 𝑘-list colorable graphs that each U𝑘LC graph is also a U(𝑘−1)LC graph [14]. That is to say that, a graph which has the property 𝑀(𝑘−1) also has the property 𝑀(𝑘).

Mahdian and Mahmoodian [5] characterized uniquely 2-list colorable graphs. They showed the following.

Proposition 1 (see [5]). A connected graph has the property 𝑀(2) if and only if every block of 𝐺 is either a cycle, a complete graph, or a complete bipartite graph.

In paper [15], it is showed that recognizing uniquely 𝑘-list colorable graphs is Σ^𝑝₂-complete for every 𝑘≥3; then uniquely 3-list colorable graphs are unlikely to have a nice characterization. But Ghebleh and Mahmoodian [14] and He et al. [16–18] have characterized the U3LC complete multipartite graphs, and one has the following.

Proposition 2 (see [14]). The graphs 𝐾_3,3,3, 𝐾_2,4,4, 𝐾_2,3,5, 𝐾_2,2,9, 𝐾_1,2,2,2, 𝐾_1,1,2,3, 𝐾_1,1,1,2,2, 𝐾_1∗4,6, 𝐾_1∗5,5, and 𝐾_1∗6,4 are U3LC.

Proposition 3 (see [16]). Let 𝐺 be a complete multipartite graph; then 𝐺 is U3LC if and only if it has one of the graphs in Proposition 2 as an induced subgraph.

Wang et al. [19] have characterized U4LC complete multipartite graphs with at least 6 parts except for finitely many of them.

In the process of characterizing U𝑘LC complete multipartite graphs, it is often researched that the property 𝑀(𝑘) of complete multipartite graphs has only one part whose size is more than one; that is, 𝐾_{1∗𝑟,𝑠}(𝑟,𝑠∈𝑁). Paper [14] studied the property 𝑀(3) of graphs 𝐾_1∗𝑟,3 and 𝐾_1,1,1,𝑟. The following was concluded.

Proposition 4 (see [14]). For every 𝑟≥2, 𝑚(𝐾_1∗𝑟,3)=𝑚(𝐾_1,1,1,𝑟)=3.

The property 𝑀(4) of graphs 𝐾_1∗𝑟,5 and 𝐾_1∗5,𝑟 is researched in paper [19], and it is showed the following.

Proposition 5 (see [19]). For every 𝑟≥1, 𝐾_1∗5,𝑟 and 𝐾_1∗𝑟,5 have the property 𝑀(4), and if 𝑟≥5, then 𝑚(𝐾_1∗5,𝑟)=𝑚(𝐾_1∗𝑟,5)=4.

Conclusions above are generalized by Wang et al. [20] recently.

Proposition 6 (see [20]). For every 𝑟≥1, 𝑘≥2, 𝐾_{1∗𝑟,(2𝑘−3)} has the property 𝑀(𝑘).

Proposition 7 (see [20]). For every 𝑟≥1, 𝑘≥2, 𝐾_{1∗(2𝑘−3),𝑟} has the property 𝑀(𝑘).

But there is no other conclusion about what are the maximal numbers 𝑟 and 𝑠 such that the graph 𝐾_{1∗𝑟,𝑠} is a U𝑘LC graph for every 𝑘. Besides, the property of list assignment of U𝑘LC graph 𝐾_{1∗𝑟,𝑠} is still unclear, and there is a lack of the necessary conditions for the U𝑘LC graph 𝐾_{1∗𝑟,𝑠}. It seems that the larger 𝑘 is, the more difficult the characterizing U𝑘LC graphs are.

In fact, if we want to proof that some graph 𝐾_{1∗𝑟,𝑠} is a U𝑘LC graph, we must find a 𝑘-list assignment such that there exists a unique list coloring. In general it is not easy to construct such list assignment, and it usually requires a lot of skills. But if some properties of such graphs are known, the construction process perhaps will become easier. In addition, it is hoped that one can obtain some properties of U𝑘LC graph 𝐾_{1∗𝑟,𝑠} for every 𝑘, not only for special 𝑘.

In this paper the property of the graph 𝐾_{1∗𝑟,𝑠} is researched when it is a U𝑘LC graph. The paper is organized as follows. In Section 2, we give some propositions about the property of the graph 𝐾_{1∗𝑟,𝑠} when it is a U𝑘LC graph. According to these propositions, a special example of U𝑘LC graphs 𝐾_{1∗𝑟,𝑠} is introduced in Section 3. In Section 4, we discuss the results and give an open problem. The conclusion will pave the way to characterize U𝑘LC complete multipartite graphs.

2. Property of the U𝑘LC Graph 𝐾_{1∗𝑟,𝑠}

In this section, we list some theorems about the property of the graph 𝐾_{1∗𝑟,𝑠} when it is U𝑘LC, as it is conducive to construct the list assignment of U𝑘LC complete multipartite graphs and characterize the U𝑘LC graphs.

Theorem 8. For every 𝑟≥1, if 𝐾_{1∗𝑟,𝑠} is a U𝑘LC graph, then 𝑠≥2 (𝑟,𝑠∈𝑁).

Proof. If 𝑠=1, 𝐾_{1∗𝑟,𝑠}=𝐾_1∗(𝑟+1) has the property 𝑀(2) by Proposition 1; so it is not a U2LC graph, nor a U𝑘LC graph which is contradictory to the suppose.

In view of these facts, it is supposed that 𝑠≥2 for a U𝑘LC graph 𝐾_{1∗𝑟,𝑠} in the following.

In the process of proving Theorem 9, for convenience, the 𝑟+1 parts of 𝐾_{1∗𝑟,𝑠} are denoted by 𝑉_𝑖={𝑣_𝑖} for 𝑖=1,2,…,𝑟 and 𝑉_𝑟+1={𝑣_𝑟+1,𝑣_𝑟+2,…,𝑣_𝑟+𝑠}. For the given 𝑘-list assignment 𝐿: 𝐿(𝑣_𝑖)={𝑐_𝑖,1,𝑐_𝑖,2,…,𝑐_𝑖,𝑘}, 𝑖=1,2,…,𝑟+𝑠, there is a unique 𝑘-list color 𝑐: 𝑐(𝑣_𝑖)=𝑐_𝑖,1, 𝑖=1,2,…,𝑟+𝑠. Furthermore, one has 𝐷_𝑖={𝑐_𝑖,1}, 𝑖=1,2,…,𝑟; 𝐷_𝑖={𝑐_𝑖,2,𝑐_𝑖,3,…,𝑐_𝑖,𝑘}, 𝑖=1,2,…,𝑟; 𝐷_𝑟+1={𝑐_𝑟+1,1,𝑐_𝑟+2,1,…,𝑐_𝑟+𝑠,1}; 𝐷_𝑟+1={𝑐_𝑟+1,2,𝑐_𝑟+1,3,…,𝑐_𝑟+1,𝑘,𝑐_𝑟+2,2,𝑐_𝑟+2,3,…,𝑐_𝑟+2,𝑘,…,𝑐_𝑟+𝑠,2,𝑐_𝑟+𝑠,3,…,𝑐_{𝑟+𝑠,𝑘}}.

Theorem 9. Suppose that 𝑟≥1, 𝑘≥3, and 𝑠≥3. Suppose that for some 𝑟, 𝑠, 𝑘, 𝐾_{1∗𝑟,𝑠} is a U𝑘LC graph, and the unique 𝑘-list color 𝑐 from the 𝑘-list assignment 𝐿 is defined as above; then one has the following:(1)𝑐_𝑖,1≠𝑐_𝑗,1, where 1≤𝑖,𝑗≤𝑟 and 𝑖≠𝑗; 𝑐_𝑖,1≠𝑐_𝑗,1, where 1≤𝑖≤𝑟, 𝑟+1≤𝑗≤𝑟+𝑠; (2)|𝐷_𝑟+1|≥2; (3)|𝐷_𝑟+1|≤𝑠−2;(4)𝑐_𝑖,1∉{𝑐_𝑗,2,𝑐_𝑗,3,…,𝑐_𝑗,𝑘}, 𝑖,𝑗=𝑟+1,𝑟+2,…,𝑟+𝑠;(5)|⋃^𝑟+1_𝑖=1𝐷_𝑖|=|⋃^𝑟+1_𝑖=1𝐷_𝑖| and 𝐷_𝑖⊆⋃_𝑗≠𝑖𝐷_𝑖, 𝑖,𝑗=1,2,…,𝑟+1; (6)there must be a 𝑖 (1≤𝑖≤𝑟) such that 𝐷_𝑖⊆𝐷_𝑟+1.

Proof. Let 𝐺=𝐾_{1∗𝑟,𝑠}.(1) From the definition of the U𝑘LC graph, this conclusion is obvious.(2) Suppose that |𝐷_𝑟+1|=1, which means that 𝑐_𝑟+1,1=𝑐_𝑟+2,1=⋯=𝑐_𝑟+𝑠,1. Let 𝐻=𝐺−𝑉_𝑟+1=𝐾_1∗𝑟. We introduce a (𝑘−1)-list assignment 𝐿^′ to 𝐻 as follows. For every vertex 𝑣 in 𝐻, if 𝑎∈𝐿(𝑣), then 𝐿^′(𝑣)=𝐿(𝑣)⧵{𝑐_𝑟+1,1}; otherwise 𝐿^′(𝑣)=𝐿(𝑣)⧵{𝑏} where 𝑏∈𝐿(𝑣) and 𝑏≠𝑐(𝑣). Since 𝐿 induces a list coloring 𝑐 for 𝐺, 𝐻 has exactly a 𝐿^′-coloring, namely, the restriction of 𝑐 on 𝐻. 𝐻=𝐾_1∗𝑟 has the property 𝑀(2) by Proposition 1; so it has the property 𝑀(𝑘), and we can obtain a new 𝐿^′-coloring of 𝐻. From the construction of 𝐿^′, we know that the new 𝐿^′-coloring can be extended to 𝐺. Thus, 𝐺 has a new 𝐿-coloring which is different from 𝑐 which is contradictory to the fact that 𝑐 is the unique 𝑘-list color.(3) We use the reduction to absurdity.
Case 1. One has |𝐷_𝑟+1|=𝑠 which means that 𝑐_𝑟+1,1,𝑐_𝑟+2,1,…,𝑐_𝑟+𝑠,1 are pairwise different.
Adding new edges between any two vertices in {𝑣_𝑟+1,𝑣_𝑟+2,…,𝑣_𝑟+𝑠}, the resulting graph is 𝐺^′=𝐾_𝑟+𝑠. Note that 𝑐 is also a proper 𝐿-coloring of 𝐺^′, and 𝐺^′ has the property 𝑀(2) by Proposition 1; hence 𝐺^′ has the property 𝑀(𝑘). So we can obtain another coloring of 𝐺^′, which is also a legal 𝐿-coloring for 𝐺, which is contradictory to the fact that 𝑐 is the unique 𝑘-list color.
Case 2. One has |𝐷_𝑟+1|=𝑠−1 which means that in {𝑣_𝑟+1,𝑣_𝑟+2,…,𝑣_𝑟+𝑠} there are just two vertices assigned a common color, and the others are pairwise different.
Not loss of generality, we say that 𝑐_𝑟+1,1=𝑐_𝑟+2,1 and 𝑐_𝑘,1 are pairwise different for 𝑘=𝑟+3,𝑟+4,…,𝑟+𝑠. Adding new edges between any two vertices in {𝑣₁,𝑣₂,…,𝑣_𝑟,𝑣_𝑟+3,𝑣_𝑟+4,…,𝑣_𝑟+𝑠}, the resulting graph is 𝐺^′′=𝐾_{𝑟+𝑠−2}. A (𝑘−1)-list assignment 𝐿^′ to 𝐺^′′ is introduced as follows. For every vertex 𝑣 in 𝐺^′′, if 𝑐_𝑟+1,1∈𝐿(𝑣), then 𝐿^′(𝑣)=𝐿(𝑣)⧵{𝑐_𝑟+1,1}; otherwise 𝐿^′(𝑣)=𝐿(𝑣)⧵{𝑏} where 𝑏∈𝐿(𝑣) and 𝑏≠𝑐(𝑣). It is obvious that |𝐿^′(𝑣)|=𝑘−1 for every 𝑣∈𝑉(𝐺^′′), and the restriction of 𝑐 on 𝐺^′′ is an 𝐿^′-coloring of 𝐺^′′. Obviously, 𝐺^′′=𝐾_{1∗(𝑟+𝑠−2)} has the property 𝑀(2) by the Proposition 1 and the property 𝑀(𝑘−1). By the property 𝑀(𝑘−1) of 𝐺^′′, we can obtain a new 𝐿^′-coloring 𝑐^′ of 𝐺^′′, which can be extended to 𝐺 as follows. For every vertex 𝑣 in 𝐺, if 𝑣∈𝑉(𝐺^′′), then 𝑐^′′(𝑣)=𝑐^′(𝑣); otherwise 𝑐^′′(𝑣)=𝑐(𝑣). From the construction of 𝐿^′, it is obvious that 𝑐^′′ is a new 𝐿-coloring of 𝐺 which is contradictory to the fact that 𝑐 is the unique 𝑘-list color.
In sum, |𝐷_𝑟+1|≤𝑠−2.(4) If 𝑖=𝑗, then it is obvious that the conclusion is true. If 𝑖≠𝑗, then we suppose that the conclusion is wrong, which means that there are two numbers 𝑖₀ and 𝑗₀ such that 𝑖₀≠𝑗₀ and 𝑐_𝑖0,1∈{𝑐_𝑗0,2,𝑐_𝑗0,3,…,𝑐_𝑗0,𝑘}. It is clear that 𝑐_𝑖0,1≠𝑐_𝑗0,1. Let 𝑐^′(𝑣_𝑗0)=𝑐_𝑖0,1 and let 𝑐^′(𝑣_𝑘)=𝑐_𝑘,1 for 𝑘=1,2,…,𝑟+𝑠 but 𝑘≠𝑗₀. Obviously, 𝑐^′ is a new 𝐿-coloring of 𝐺 which is contradictory to the fact that 𝑐 is the unique 𝑘-list color.(5) Proof by contradiction. Suppose that |⋃^𝑟+1_𝑖=1𝐷_𝑖|≠|⋃^𝑟+1_𝑖=1𝐷_𝑖|, and there are 𝑖₀ and 𝑗₀ such that 1≤𝑖₀≤𝑟+𝑠, 2≤𝑗₀≤𝑘 and 𝑐_𝑖0,𝑗0∉⋃^𝑟+1_𝑖=1𝐷_𝑖. Let 𝑐^′(𝑣_𝑖0)=𝑐_𝑖0,𝑗0 and let 𝑐^′(𝑣_𝑘)=𝑐_𝑘,1 for 𝑘=1,2,…,𝑟+𝑠 but 𝑘≠𝑖₀. Obviously 𝑐^′ is a new 𝐿-coloring of 𝐺 which is contradictory to the fact that 𝑐 is the unique 𝑘-list color. Then from (3) we know that 𝐷_𝑖⊆⋃_𝑗≠𝑖𝐷_𝑖, 𝑖,𝑗=1,2,…,𝑟+1.(6) By contradiction. Suppose for every 𝑖 (1≤𝑖≤𝑟) that 𝐷_𝑖⊆𝐷̸_𝑟+1; then it is obtained that |𝐷_𝑖⧵𝐷_𝑟+1|≥1. So we get that |𝐿(𝑣_𝑖)⧵𝐷_𝑟+1|≥2 for every 1≤𝑖≤𝑟. Let 𝐻=𝐺−𝑉_𝑟+1=𝐾_1∗𝑟. We introduce a 2-list assignment 𝐿^′ to 𝐻 as follows. For every vertex 𝑣_𝑖 in 𝐻, we obtain 𝐿^′(𝑣_𝑖) by randomly getting rid of 𝑘−2 elements from 𝐿(𝑣_𝑖) such that 𝐿^′(𝑣_𝑖)∩𝐷_𝑟+1=Φ and |𝐿^′(𝑣_𝑖)|=2, as can be done because |𝐿(𝑣_𝑖)⧵𝐷_𝑟+1|≥2. Since 𝐿 induces a list coloring 𝑐 for 𝐺, 𝐻 has exactly one 𝐿^′-coloring, namely, the restriction of 𝑐 on 𝐻. 𝐻=𝐾_1∗𝑟 has the property 𝑀(2) by Proposition 1; so we can obtain a new 𝐿^′-coloring of 𝐻. From the construction of 𝐿^′, we know that the new 𝐿^′-coloring can be extended to 𝐺. Thus, 𝐺 has a new 𝐿-coloring which is different from 𝑐 which is contradictory to the fact that 𝑐 is the unique 𝑘-list color.

3. An Example of U𝑘LC Graphs 𝐾_{1∗𝑟,𝑠}

According the property of U𝑘LC graph in Theorem 9, we construct a list assignment of graph 𝐾_{1∗𝑟,𝑠} for special 𝑟 and 𝑠 and prove that the graph is a U𝑘LC graph in this section.

Theorem 10. Let 𝑀=(𝑘−1)^2𝑘−2_𝑘−1 where 𝑘≥2, the graph 𝐾_{1∗(2𝑘−2),𝑀} is a U𝑘LC graph.

Proof. For convenience, the (2𝑘−1) parts of 𝐾_{1∗(2𝑘−2),𝑀} are denoted by 𝑉_𝑖={𝑣_𝑖} for 𝑖=1,2,…,2𝑘−2 and 𝑉_2𝑘−1={𝑢₁,𝑢₂,…,𝑢_𝑀}.
Let 𝐴={1,2,…,2𝑘−2}. We denote all (𝑘−1)-subsets of 𝐴 by {𝐴₁,𝐴₂,…,𝐴_𝑚}, where 𝑚=^2𝑘−2_𝑘−1. Now consider 𝐾_{1∗(2𝑘−2),𝑀} with the following 𝑘-list of colors on vertices. For 𝑖=1,2,…,2𝑘−2, 𝐿(𝑣_𝑖)={𝑖,2𝑘−1,2𝑘,…,3𝑘−3}. For every 𝑡=1,2,…,𝑚, 𝑗=1,2,…,𝑘−1, 𝐿(𝑢_{(𝑗−1)∗𝑚+𝑡})={𝑗+2𝑘−2}∪𝐴_𝑡.
For example, when 𝑘=3, 𝐾_{1∗(2𝑘−2),𝑀}=𝐾_1∗4,12, and the above mentioned 𝑘-list assignment 𝐿 for 𝐾_1∗4,12 is as follows: 156,256,356,4,55612,513,514,523,524,534,612,613,614,623,624,634. (1)
Note that the list assignment 𝐿 makes a total of 3𝑘−3 colors. Since 𝐾_{1∗(2𝑘−2),𝑀} is a complete (2𝑘−1)-partite graph, the last part 𝑉_2𝑘−1 can take 𝑘−1 colors at the most. From the construction of 𝐿, it is obtained obviously that {2𝑘−1,2𝑘,…,3𝑘−3} is the unique choice for a 𝑘-list coloring from 𝐿. Then the 𝑉_𝑖(1≤𝑖≤2𝑘−2) must take the color 𝑖. In the example above, the colors in the list coloring are marked by underlines. So a unique 𝑘-list coloring from 𝐿 is made and 𝐾_{1∗(2𝑘−2),𝑀} is a U𝑘LC graph.

Notice that for every 𝑟≥1, 𝑘≥2, 𝐾_{1∗(2𝑘−3),𝑟} has the property 𝑀(𝑘) by Proposition 7. Now the graph 𝐾_{1∗(2𝑘−2),𝑀} in Theorem 10 is a U𝑘LC graph; so it is wrong with the proposition “for every 𝑟≥1, 𝑘≥2, 𝐾_{1∗(2𝑘−2),𝑟} has the property 𝑀(𝑘)”. Therefore, (2𝑘−3) is the maximal numbers in Proposition 7.

4. Discussion and Some Open Problems

It is not easy to characterize U𝑘LC complete multipartite for any 𝑘. In fact, it is a very tricky job to construct a 𝑘-list assignment such that there exists a unique list coloring. Theorem 9 provides a direction for constructing such list assignment of 𝐾_{1∗𝑟,𝑠}, and perhaps it makes construction easier for the researchers. Furthermore, Theorem 9 is true for every 𝑘(𝑘≥2 and 𝑘∈𝑁), and the conclusion is extensive.

It must be noted that the conditions in Theorem 9 are only necessary conditions of 𝐾_{1∗𝑟,𝑠} for U𝑘LC graph, not sufficient conditions.

Theorem 10 can be regarded as a application of Theorem 9. And from Theorem 10, it is known that (2𝑘−3) is exactly the maximal numbers in Proposition 7. But notice that the number 𝑀=(𝑘−1)^2𝑘−2_𝑘−1 is not the minimal number for every 𝑘 in Proposition 7. For example, when 𝑘=3, 𝑀=12 and 𝐾_1∗4,12 is U3LC according to Theorem 10. In fact 𝐾_1∗4,6 is U3LC by Propositions 2 and 3; so 𝑀=12 is not the minimal number for 𝑘=3 in Proposition 7. Moreover, it is very likely that for different 𝑘 the minimal number is different in Proposition 7.

The following problem arises naturally from the work.

Problem. For every 𝑘, characterize all minimum number 𝑠 such that the graph 𝐾_{1∗(2𝑘−2),𝑠} is a U𝑘LC graph.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (No. 61271409), the China Postdoctoral Science Foundation (No. 2012M510768, No. 2013T60264), and a grant from the Science and Technology Research and Development Program of Qinhuangdao (No. 201001A151).