On ( 2 , t )-Choosability of Triangle-Free Graphs

A (k, t)-list assignment L of a graph G is a mapping which assigns a set of size k to each vertex v of G and |⋃v∈V(G) L(v)| = t. A graph G is (k, t)-choosable if G has a proper coloring f such that f (v) ∈ L(v) for each (k, t)-list assignment L. In 2011, Charoenpanitseri, Punnim and Uiyyasathian proved that every n-vertex graph is (2, t)-choosable for t ≥ 2n − 3 and every n-vertex graph containing a triangle is not (2, t)-choosability for t ≤ 2n − 4. Then a complete result on (2, t)-choosability of an n-vertex graph containing a triangle is revealed. Moreover, they showed that an n-vertex triangle-free graph is (2, t)-choosable for t ≥ 2n − 6. In this paper, we first prove that an n-vertex graph containing K3,3 − e is not (2, t)-choosable for t ≤ 2n − 7. Then we deeply investigates (2, t)-choosablity of an n-vertex graph containing neither a triangle nor K3,3 − e.


Introduction
A k-list assignment L of a graph G is a mapping which assigns a set of size k to each vertex v of G.A (k, t)-list assignment of G is a k-list assignment with | v∈V(G) L(v)| = t.Given a list assignment L, a proper coloring f of G is an L-coloring of G if f (v) is chosen from L(v) for every vertex v of G.A graph G is L-colorable if G has an L-coloring.Particularly, if L is a (k, k)-list assignment of G, then any L-coloring of G is a k-coloring of G.A graph G is (k, t)-choosable if G is L-colorable for every (k, t)-list assignment L. If a graph G is (k, t)-choosable for each positive number t then G is called k-choosable.
List coloring is a well-known problem in the field of graph theory.It was first studied by Vizing (1976) and by Erdős, Rubin and Taylor (1979).They give a characterization of 2-choosable graphs.Recall that a property of 2choosable graphs is that all vertices can be colored under the condition every adjacent vertices is labeled by distinct colors whenever the vertices have exactly two available colors.To prove that a graph is k-choosable, we need to prove that the graph can be colored for all k-list assignments.Hence, the problem is quite complicated because of a large number of k-list assignments.For k ≥ 3, there is no characterization of k-choosable graphs.There are only results for some classes of graphs.For example, all planar graphs are 5-choosable, while some planar graphs are 3-choosable (See Lam, Shiu, & Song, 2005;Thomassen, 1995;Thomassen, 1994;Zhang, 2005;Zhang & Xu, 2004;Zhang, Xu, & Sun, 2006;Zhu, Lianying, & Wang, 2007).
In order to simplify the problem, (k, t)-choosability is defined.It is a partial problem of k-choosability.Instead of proving a graph can always be colored for entire k-list assignments, we prove the graph can be colored for k-list assignments that have exactly t colors.For example, Ganjari et al. (2002) apply (k, t)-choosability of graphs to generalize the characterization of uniquely 2-list colorable graphs.Recently, (k, t)-choosability of graphs is explored in Charoenpanitseri, Punnim, and Uiyyasathian (2011).They prove that every n-vertex graph is (k, t)choosable for t ≥ kn − k 2 + 1 and every n-vertex graph containing K k+1 is not (k, t)-choosable for t ≤ kn − k 2 .In case k = 2, they prove that every n-vertex graph is (2, t)-choosable for t ≥ 2n − 3 and every n-vertex graph containing a triangle is not (2, t)-choosable for t ≤ 2n − 4.Moreover, every n-vertex graph not containing a triangle is (2, t)-choosable for t ≥ 2n − 6.
Second, an n-vertex graph containing neither a triangle nor K 3,3 − e is (2, t)-choosable for t ≥ 2n − 7. Third, an n-vertex graph containing C 5 or a domino is not (2, t)-choosable for 3 ≤ t ≤ 2n − 8.Last but not least, an n-vertex graph containing neither a triangle, K 3,3 − e, a domino, Throughout the paper, G denotes a simple, undirected, finite, connected graph; V(G) and E(G) are the vertex set and the edge set of G.A cycle is a graph with an equal number of vertices and edges whose vertices can be placed around a circle so that two vertices are adjacent if and only if they appear consecutively along the circle; the cycle with n vertices is denoted by C n .A complete graph is a graph whose vertices are pairwise adjacent; the complete graph with n vertices is denoted by K n .A triangle is a complete graph with 3 vertices.A graph G is bipartite if V(G) is the union of two disjoint independent sets called partite sets.A complete bipartite graph is a bipartite graph such that two vertices are adjacent if and only if they are in different partite sets; the complete bipartite graph with partite sets of size a and b is denoted by K a,b .The graph obtained from deleting an edge from the graph K 3,3 is denoted by K 3,3 − e.
The subgraph induced by X, denoted by G[X] is the graph obtained from deleting all vertices of V(G) outside X.
When t < k or t > kn, there is no (k, t)-list assignment, so it is automatically (k, t)-choosable.Unless we say otherwise, our parameters k, n and t in this paper are always numbers such that t ≥ k.If k ≥ n then all of the n-vertex graphs are (k, t)-choosable.
Example 1.1 A bipartite graph K 3,3 − e is not (2, t)-choosable for t = 3, 4, 5.If u 1 and u 2 are labeled by color 1, the vertex v 3 cannot labeled.If u 1 or u 2 is labeled by color 1, then v 1 and v 2 must be labeled by color a and b, respectively.Consequently, the vertex u 3 cannot be labeled.Hence, K 3,3 − e is not L-colorable.Therefore, K 3,3 − e is not (2, t)-choosable for t = 3, 4, 5.
Proof.Suppose t = 3 or 4. Let L be a (2, t)-list assignment of a domino with the vertex set v 1 , v 2 , . . ., v 6 as shown in the figure.If v 2 is labeled by color 1, then v 3 and v 5 must be labeled by color 3 and color 2, respectively.Hence v 4 cannot be labeled.If v 2 is labeled by color 2, then v 1 and v 5 must be labeled by color a and color 1, respectively.Hence v 6 cannot be labeled.That is, the domino is not L-colorable.Therefore, G is not (2, t)-choosable for t = 3, 4.
In 2011, Charoenpanitseri, Punnim and Uiyyasathian give a complete result on (k, t)-choosablity of an n-vertex graph containing K k+1 .Particulary, a complete result on (2, t)-choosability of an n-vertex graph containing a triangle is revealed as shown in Theorem 1.3.Before going to our main results, we will introduce some tools using in our proof.Theorem 1.4 and Lemma 1.5 are applied when we prove that a graph is (2, t)-choosable for some number t while Lemma 1.6 is applied when we prove that a graph is not (2, t)-choosable for some number t.
Let S ⊆ V(G).If L is a list assignment of G, we let L| S denote L restricted to S and L(S ) denote v∈S L(v).
Theorem 1.4 (Kierstead, 2000) Let L be a list assignment of a graph G and let S ⊆ V(G) be such that Proof.Let H be an m-vertex subgraph of an n-vertex graph G. Let t 0 , t be numbers such that t 0 ≤ t ≤ 2n − 2m + t 0 .Assume that H is not (2, t 0 )-choosable.Hence, there is a (2, t 0 )-list assignment L 0 such that H is not L 0 -colorable.Then we extend a (2, t 0 )-list assignment L 0 of H to a (2, t)-list assignment L of G by assigning the remaining colors to the remaining vertices outside V(H).Notice that G has n − m remaining vertices and L has t − t 0 remaining colors.The condition t − t 0 ≤ 2n − 2m can confirm the existence of L. Since H is not L 0 -colorable, G is not L-colorable.Consequently, G is not (2, t)-choosable.

Main Results
In Charoenpanitseri et al. (2011), the authors show that an n-vertex graph not containing a triangle is (2, t)choosable for t ≥ 2n − 6.Then we study (2, t)-choosability of a triangle-free graph when t ≤ 2n − 7. The first result is that an n-vertex graph containing Next, we focus on an n-vertex graph containing neither a triangle nor K 3,3 − e.Let us introduce a theorem on (2, t)-choosability of a triangle-free graph.
Theorem 2.2 (Charoenpanitseri et al., 2011) A triangle-free graph with n vertices is (2, 2n − 7)-choosable if and only if it does not contain K 3,3 − e as a subgraph.
We apply the above theorem to obtain the second result in Theorem 2.3.
Proof.Let G be an n-vertex graph containing neither a triangle nor Now, the result in case t ≥ 2n − 7 is revealed.Then we keep studying in the remaining case; the case that t ≤ 2n − 8.The third result is that every n-vertex graph containing a domino,  If there is a color c that appears in only one vertex, then we can label the vertex by color c and the remaining vertices can be labeled.If there is an edge e that endpoints has no common color, then G[S ] − e is easily be colored.Suppose that each color appears in at least 2 vertices and endpoints of each edge share a common color.Since G[S ] has 7 vertices and 6 colors, a color appears in 4 vertices and 5 colors appear in 2 vertices, or 2 colors appear in 3 vertices and 4 colors appear in 2 vertices.
Case 3.4.2.x 1 and x 3 has a common color.The proof is similar to Case 3.4.1.

Applications
In this section, we apply our main results to some classes of graphs such as grid graphs and hypercube graphs.We start this section with definitions and examples of the two classes of graphs.
A grid graph is a unit distance graph corresponding to the square lattice, so that it is isomorphic to the graph having a vertex corresponding to every pair of integers (a, b), and an edge connecting    Proof.Let G be an n-vertex triangle-free and K 3,3 − e-free graph containing a domino and t ≥ 3.

Figure 2 .
Figure 2. A 2-list assignment L of a domino where a is color 3 or color 4 Theorem 1.3 (Charoenpanitseri et al., 2011)  Let G be an n-vertex graph.If G contains a triangle, then it is not (2, t)-choosable for t ≤ 2n − 4. If G does not contain a triangle, then it is (2, t)-choosable for t ≥ 2n − 6.

Figure 3 .
Figure 3.The graph C4 • C4 and its (2, 6)-list assignment Proof.Let L be (2, 6)-list assignment as shown in Figure 3. Suppose that C 4 • C 4 is L-colorable.If v 3 is labeled by color 1, then v 2 and v 4 must be labeled by color 3 and color 4, respectively.Hence, v 1 has no available color; a contradiction.If v 3 is labeled by color 2, then v 5 and v 7 must be labeled by color 5 and color 6, respectively.Hence, v 6 has no available color; a contradiction.Theorem 2.4 An n-vertex graph containing a domino, C 5 , K 2,4 or C 4 • C 4 is not (2, t)-choosable for 3 ≤ t ≤ 2n − 8.Proof.Let G be an n-vertex graph.

Figure 4 .
Figure 4.A subgraph of K 3,4 (a, b) to (a + 1, b) and (a, b + 1).The finite grid graph G(m, n) is an m × n rectangular graph isomorphic to the one obtained by restricting the ordered pairs to the range 0 ≤ a < m, 0 ≤ b < n.A domino is G(2, 3) (See examples in Figure 5).

Figure 5 .
Figure 5. Examples of grid graphs An a-hypercube graph, denoted by Q a , is the graph whose vertices are the a tuples with entries in {0, 1} and whose edges are the pair of a-tuples that differ in exactly one position (See examples in Figure 6).

Figure 6 .
Figure 6.Examples of hypercubes According to the four main result, (2, t)-choosability of some classes of graphs are obtained.