Bipartite graphs whose squares are not chromatic-choosable

The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Let $\chi(H)$ and $\chi_{\ell}(H)$ be the chromatic number and the list chromatic number of $H$, respectively. A graph $H$ is called {\em chromatic-choosable} if $\chi_{\ell} (H) = \chi(H)$. It is an interesting problem to find graphs that are chromatic-choosable. Motivated by the List Total Coloring Conjecture, Kostochka and Woodall (2001) proposed the List Square Coloring Conjecture which states that $G^2$ is chromatic-choosable for every graph $G$. Recently, Kim and Park showed that the List Square Coloring Conjecture does not hold in general by finding a family of graphs whose squares are complete multipartite graphs with partite sets of unbounded size. It is a well-known fact that the List Total Coloring Conjecture is true if the List Square Coloring Conjecture holds for special class of bipartite graphs. On the other hand, the counterexamples to the List Square Coloring Conjecture are not bipartite graphs. Hence a natural question is whether $G^2$ is chromatic-choosable or not for every bipartite graph $G$. In this paper, we give a bipartite graph $G$ such that $\chi_{\ell} (G^2) \neq \chi(G^2)$. Moreover, we show that the value $\chi_{\ell}(G^2) - \chi(G^2)$ can be arbitrarily large.

From now on, we fix a prime number n with n ≥ 3. For i ∈ [n − 1], we define a Latin square L i of order n by L i (j, k) = j + i(k − 1) (mod n), for (j, k) ∈ [n] × [n]. (2.1) Then it is easily checked (and well-known) that L i is a Latin square of order n and {L 1 , L 2 , . . . , L n−1 } is a family of mutually orthogonal Latin squares of order n as n is prime (see page 252 in [8]). In Figure 1, L 1 and L 2 are orthogonal Latin squares defined in (2.1) for n = 3. Now we will construct a bipartite graph G such that G 2 is not chromatic-choosable. First, we will describe briefly how to construct such bipartite graph G, and then will give a formal description in Construction 2.1.
The procedure of the construction of G Step 1: For each prime number n ≥ 3, we construct a graph H n with 2n 2 vertices as follows. For k ∈ [n], let P k be the set of n elements such that P k = {v k,1 , v k,2 , ..., v k,n }, and for i ∈ [n − 1], let Q i be the set of n elements such that Q i = {w i,1 , w i,2 , ...., w i,n }, and let S be the set of n elements such that S = {s 1 , s 2 , ...., s n }.
Let {L 1 , L 2 , . . . , L n−1 } be the family of mutually orthogonal Latin squares of order n obtained by (2.1). Graph H n is defined as follows: {s j v k,j : k ∈ [n]}   . . Figure 2: Graph H n when n = 3 in Step 1. The bold edges induce the graph obtained by removing the edges in j∈ [3] {xy : x, y ∈ T j } from graph G 3 in [2].
and H n is the graph obtained by removing the edges in j∈ [n] {xy : x, y ∈ T j } from the graph G n in [2] and adding vertices of S and the edges of j∈[n] {xs j : x ∈ T j }. (Figure 2 is the case when n = 3.) Given a graph H and a vertex v in H, duplicating v means adding a new vertex v 0 and making it adjacent to all the neighbors of v in H, but v and v 0 are not adjacent. (See Figure 3 for an illustration.) Step 2: Duplicate each vertex of ∪ n k=1 P k exactly (n − 1) times. For each vertex v k,j , denote the (n − 1) copies of v k,j by v 2 k,j , v 3 k,j , . . . , v n k,j , and denote the original vertex v k,j by v 1 k,j .
Let Step 3: ..., u i,n }. For each vertex u i,j , make u i,j adjacent to all vertices in ∪ n l=1 T l,L i (j,l) . Note that the neighborhood of u i,j follows the same pattern of the neighborhood of w i,j . For example, if N Hn (w i,j ) = {v k,L i (j,k) : k ∈ [n]}, then N G (u i,j ) = ∪ n k=1 T k,L i (j,k) . Now, call the resulting graph by G. Figure 4 is an illustration of G when n = 3, and its description is below. Figure 4: For each l ∈ [3], the dotted line abbreviates adjacency between P l 1 ∪ P l 2 ∪ P l 3 and Q 1 ∪ Q 2 . For each l ∈ [3], the bold line abbreviates adjacency between P l 1 ∪ P l 2 ∪ P l 3 and R 1 ∪ R 2 , and the doubled thin line abbreviates adjacency between the union of all P l k 's and S. In N G (w i,j ) and N G (u i,j ), the bold subscripts are the jth row of the Latin square L i which was defined in Figure 1, respectively. Note that for each l ∈ [3] the subgrah induced by P l Figure 2.

Description of
The following is a formal description of the construction of G.
Construction 2.1. We construct a graph G with n(n 2 + 2n − 1) vertices as follows. For each k, l ∈ [n], let P l k be the set of n elements such that Now we define a graph G as follows:  where {s m y : y ∈ T l,m for some l ∈ [n]}.
By the definition of the graph G, it follows that For simplicity, for each l ∈ [n], let P l = P l 1 ∪ · · · ∪ P l n and let Let K n * r denote the complete multipartite graph with r partite sets in which each partite set has n vertices. We will show that the subgraph of G 2 induced by P is the complete multipartite graph K n⋆n 2 whose partite sets are {P l k : k, l ∈ [n]}. For each l ∈ [n], let G l be the subgraph of G induced by P l ∪ Q. The following properties were obtained in Lemma 2.2 and Lemma 2.4 in [2]. (1) For any vertex w ∈ Q, (2) For any distinct vertices w and w ′ in Q, (3) For any vertex w ∈ Q, (4) For any vertex v ∈ P l , From Lemma 2.2, we show the following lemmas. (3) The set S is an independent set of G 2 .
Proof. Consider P l k for some k, l ∈ [n]. Let v, v ′ be distinct vertices in P l k . We will show that v and v ′ do not have a common neighbor. First, the vertices v and v ′ do not have a common neighbor in Q by (1) of Lemma 2.2. Next, |N G (u) ∩ P l k | = 1 for any u ∈ R by (2.3), and |N G (s) ∩ P l k | = 1 for any s ∈ S by (2.4). Hence v and v ′ do not have a common neighbor in R ∪ S. Thus v and v ′ do not have a common neighbor in G, and consecutively, v and v ′ are not adjacent in G 2 . Therefore, P l k is an independent set in G 2 .
Let w and w ′ be any distinct vertices in Q i . Suppose that the vertices w and w ′ are adjacent in G 2 . Since G is a bipartite graph, they have a common neighbor v in P . Then v ∈ P l for some l. Thus w, w ′ ∈ N G l (v) ∩ Q i . But, by (4) of Lemma 2.2, |N G l (v)∩Q i | = 1. This is a contradiction for the assumption that w and w ′ are distinct. Therefore for each i ∈ [n − 1], Q i is an independent set in G 2 .
Let u = u i,j and u ′ = u i,j ′ be any distinct vertices in R i . Suppose that the vertices u and u ′ are adjacent in G 2 . Then they have a common neighbor v in P . Then v ∈ N G (u)∩N G (u ′ ), and so by (2.3), v ∈ T a,L i (j,a) ∩T b,L i (j ′ ,b) for some a and b. Therefore, a = b and L i (j, a) = L i (j ′ , a), which implies j = j ′ since L i is a Latin square. This is a contradiction. Thus for each i ∈ [n − 1], R i is an independent set in G 2 .
Moreover, it is clear that by (2.4), any two vertices in S do not have a common neighbor in G, and so S is an independent set of G 2 .
Let G 2 [P l ] denote the subgraph of G 2 induced by P l . Lemma 2.4. For each l ∈ [n], G 2 [P l ] ∼ = K n * n whose partite sets are P l 1 , P l 2 ,. . . , P l n .
Proof. The proof of Lemma 2.4 is basically similar to the proof of Lemma 2.8 in [2], but we will include here for the sake of completeness. Take an integer l ∈ [n]. Note that N G l (w) ⊂ P l for each w ∈ Q by the definition of G l . First, note that G 2 [P l ] is isomorphic to a subgraph of K n⋆n , since for each k, l ∈ [n], P l k is an independent set of G 2 [P l ] by Lemma 2.3. Let Note that for each w ∈ Q, N G (w) ∩ P l induces a complete graph K n in G 2 , and for each s ∈ S, N G (s) ∩ P l induces a complete graph K n in G 2 . Therefore F l is a family of copies of K n .
For any two vertices w, w ′ ∈ Q, we have are edge-disjoint. Therefore any two cliques in F l are edge-disjoint.
In addition, |F l | = |Q| + |S| = n(n − 1) + n = n 2 . Thus F l is a family of n 2 pairwise edge-disjoint cliques of size n in G 2 [P l ]. It follows that Hence G 2 [P l ] ∼ = K n * n for each l ∈ [n], since G 2 [P l ] is isomorphic to a subgraph of K n * n .
To show that G 2 [P ] ∼ = K n⋆n 2 , it is remained to show the following lemma.
Lemma 2.5. For any distinct s, t ∈ [n], for any v ∈ P s and v ′ ∈ P t , v and v ′ are adjacent in G 2 .
Proof. Let v ∈ P s and v ′ ∈ P t be any vertices with s = t. Then v ∈ T s,a and v ′ ∈ T t,b for some a, b ∈ [n]. If a = b, then T s,a ∪ T t,b ⊂ N G (s a ) and so v and v ′ have a common neighbor s a in G. Hence vv ′ ∈ E(G 2 ). We will show that if a = b, then there exist i ∈ [n − 1] and j ∈ [n] such that L i (j, s) = a and L i (j, t) = b for fixed s and t. Note that if a = b and s = t, then there exist i and j satisfying the following equations.
Thus from (2.1), we know that there exist i ∈ [n − 1] with L i (j, s) = a since a = b, and , and so v and v ′ have a common neighbor u i,j in G.
Theorem 2.6. If G be the graph defined in Construction 2.1, then G 2 [P ] ∼ = K n⋆n 2 whose partite sets are P l k 's.
The following lower bound on the list chromatic number of a complete multipartite graph was obtained in [9]. Theorem 2.7. (Theorem 4, [9]) For a complete multipartite graph K n * r with n, r ≥ 2, Consequently, we obtain that χ ℓ (G) > χ(G) by the following theorem.

Further Discussion
Note that from (4) of Lemma 2.2, each vertex v in P has exactly one neighbor in each of Q i , R j , and S, respectively. Thus, if G is the bipartite graph defined in Construction 2.1 for prime number n, then d G (x) = 2n − 1 for each x ∈ P and d G (y) = n 2 for each y ∈ Q ∪ R ∪ S. Hence from Theorem 2.8, if G is the bipartite graph defined in Construction 2.1 for n = 7, then G 2 is not chromatic-choosable and every vertex of one partite set of G has degree 13. Note that the List Total Coloring Conjecture is true if the List Square Coloring Conjecture holds for bipartite graphs such that every vertex of one partite set has degree at most 2. Thus, it would be interesting to answer the following questions.
Question 3.1. If G is a bipartite graph such that every vertex of one partite set has degree at most 2, then is it true that χ ℓ (G 2 ) = χ(G 2 )?
Question 3.2. If Question 3.1 is true, then what is the largest k such that G 2 is chromatic-choosable for every bipartite graph G with a partite set in which each vertex has degree at most k?
We already mentioned that there is a bipartite graph G such that every vertex of one partite set of G has degree 13 and G 2 is not chromatic-choosable. Thus if Question 3.1 is true (or the List Total Coloring Conjecture is true), then the k in Question 3.2 must be less than 13.
On the other hand, if we apply to the 'duplication idea' in Step 2 in the procedure of the construction of G repeatedly, then we can obtain a bipartite graph G such that every vertex of one partite set of G has degree 7 and G 2 is not chromatic-choosable. This implies that the integer k in Question 3.2 must be less than 7.
We will describe briefly how to construct such bipartite graph G. Let G be the graph in Construction 2.1 when n = 3. Now we duplicate each vertex of P exactly 2 times. For each vertex v l k,j , we denote its copies by v ′ l k,j and v ′′ l k,j . Let P ′ denote the set of the first copied vertices v ′ l k,j , and let P ′′ denote the set of the second copied vertices v ′′ l k,j . For each h ∈ [3], let T 1,h = T 1,h ∪ T 2,h ∪ T 3,h , that is, In addition, let the two copies corresponding to T 1,h be denoted as follows: Next, we introduce 6 new vertices of B 1 ∪ B 2 where B 1 = {b 1,1 , b 1,2 , b 1,3 } and B 2 = {b 2,1 , b 2,2 , b 2,3 }, in which the neighborhood of each vertex b i,j ∈ B 1 ∪ B 2 follows the same pattern of the neighborhood of w i,j (similar to Step 3 in the procedure of the construction of G). More precisely, N G (b i,j ) = ∪ 3 k=1 T k,L i (j,k) for each b i,j , where G is the resulting graph. See Figure 5 for an illustration and its description is below.
Description of Figure 5: The sets P ′ and P ′′ are copies of P , and the bold line abbreviates adjacency between P and S ∪ Q ∪ R. Each of three P ∪ S ∪ Q ∪ R, P ′ ∪ S ∪ Q ∪ R, P ′′ ∪ S ∪ Q ∪ R induces a graph isomorphic to graph G in Figure 4. Like as N G (w i,j ) and N G (u i,j ), in N G (b i,j ), the bold subscripts are the jth row of the Latin square L i which was defined in Figure 1.
Then, the resulting graph G is a bipartite graph with partite sets X = P ∪ P ′ ∪ P ′′ and Y = S ∪Q∪R∪B 1 ∪B 2 . Note that each vertex x ∈ X has degree 7 and each vertex y ∈ Y has degree 27. Then for each k, l ∈ [3], we can see that P l k = {v l k,1 , v l k,2 , v l k,3 , } is an independent set in G 2 and each of its corresponding copies P ′ l k and P ′′ l k is also an independent set in G 2 . In addition, each of S, Q 1 , Q 2 , R 1 , R 2 , B 1 , B 2 is an independent Thus we know that G 2 is a multipartite graph with 34 partite sets. Therefore χ(G 2 ) ≤ 34. Moreover, we can easily check that the subgraph of G 2 induced by X is the complete multipartite graph K 3⋆27 , and so χ ℓ (G 2 ) ≥ χ ℓ (K 3⋆27 ) = ⌈ 4×27−1 3 ⌉ = 36 (see [5]). Thus G 2 is not chromatic-choosable. Remark 3.3. In general, for each prime number n, if we apply this duplication idea d times to the graph H n in the Construction 2.1, then we have a bipartite graph whose square is a multiparite graph with n d + d(n − 1) + 1 partite sets, containing a complete multiparitite graph K n⋆n d . Through this way, we can also construct many bipartite graphs whose square are not chromatic-choosable.