Abstract:
It is known to be a hard problem to compute the competition number k(G) of a graph G in general. Park and Sano [13] gave the exact values of the competition numbers of Hamming graphs H(n,q) if $1 \leq n \leq 3$ or $1 \leq q \leq 2$. In this paper, we give an explicit formula of the competition numbers of ternary Hamming graphs.

Abstract:
The competition graph of a digraph D is a graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x,v) and (y,v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is defined to be the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of important research problems in the study of competition graphs. In this paper, we compute the competition numbers of Hamming graphs with diameter at most three.

Abstract:
The competition graph of a doubly partial order is known to be an interval graph. The CCE graph and the niche graph of a doubly partial order are also known to be interval graphs if the graphs do not contain a cycle of length four and three as an induced subgraph, respectively. Phylogeny graphs are variant of competition graphs. The phylogeny graph $P(D)$ of a digraph $D$ is the (simple undirected) graph defined by $V(P(D)):=V(D)$ and $E(P(D)):=\{xy \mid N^+_D(x) \cap N^+_D(y) \neq \emptyset \} \cup \{xy \mid (x,y) \in A(D) \}$, where $N^+_D(x):=\{v \in V(D) \mid (x,v) \in A(D)\}$. In this note, we show that the phylogeny graph of a doubly partial order is an interval graph. We also show that, for any interval graph $G$, there exists an interval graph $\tilde{G}$ such that $\tilde{G}$ contains the graph $G$ as an induced subgraph and that $\tilde{G}$ is the phylogeny graph of a doubly partial order.

Abstract:
In 1982, Opsut showed that the competition number of a line graph is at most two and gave a necessary and sufficient condition for the competition number of a line graph being one. In this note, we generalize this result to the competition numbers of generalized line graphs, that is, we show that the competition number of a generalized line graph is at most two, and give necessary conditions and sufficient conditions for the competition number of a generalized line graph being one.

Abstract:
The competition hypergraph $C{\cH}(D)$ of a digraph $D$ is the hypergraph such that the vertex set is the same as $D$ and $e \subseteq V(D)$ is a hyperedge if and only if $e$ contains at least 2 vertices and $e$ coincides with the in-neighborhood of some vertex $v$ in the digraph $D$. Any hypergraph with sufficiently many isolated vertices is the competition hypergraph of an acyclic digraph. The hypercompetition number $hk(\cH)$ of a hypergraph $\cH$ is defined to be the smallest number of such isolated vertices. In this paper, we study the hypercompetition numbers of hypergraphs. First, we give two lower bounds for the hypercompetition numbers which hold for any hypergraphs. And then, by using these results, we give the exact hypercompetition numbers for some family of uniform hypergraphs. In particular, we give the exact value of the hypercompetition number of a connected graph.

Abstract:
Given a simple graph $G$, the graph associahedron $P_G$ is a convex polytope whose facets are corresponding to the connected induced subgraphs of $G$. Graph associahedra have been studied widely and founded in a broad range of subjects. Recently, Choi and Park computed the rational Betti numbers of the real toric variety corresponding to a graph associahedron under the canonical Delzant realization. In this paper, we focus on a pseudograph associahedron which was introduced by Carr, Devadoss and Forcey, and then discuss how to compute the Poincar\'{e} polynomial of the real toric variety corresponding to a pseudograph under the canonical Delzant realization.

Abstract:
The Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-element subsets of an $n$ elements set, with two vertices adjacent if they are disjoint. 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. Determining the chromatic number of the square of the Kneser graph $K(n, k)$ is an interesting graph coloring problem, and is also related with intersecting family problem. The square of $K(2k, k)$ is a perfect matching and the square of $K(n, k)$ is the complete graph when $n \geq 3k-1$. Hence coloring of the square of $K(2k +1, k)$ has been studied as the first nontrivial case. In this paper, we focus on the question of determining $\chi(K^2(2k+r,k))$ for $r \geq 2$. Recently, Kim and Park \cite{KP2014} showed that $\chi(K^2(2k+1,k)) \leq 2k+2$ if $ 2k +1 = 2^t -1$ for some positive integer $t$. In this paper, we generalize the result by showing that for any integer $r$ with $1 \leq r \leq k -2$, (a) $\chi(K^2 (2k+r, k)) \leq (2k+r)^r$, if $2k + r = 2^t$ for some integer $t$, and (b) $\chi(K^2 (2k+r, k)) \leq (2k+r+1)^r$, if $2k + r = 2^t-1$ for some integer $t$. On the other hand, it was showed in \cite{KP2014} that $\chi(K^2 (2k+r, k)) \leq (r+2)(3k + \frac{3r+3}{2})^r$ for $2 \leq r \leq k-2$. We improve these bounds by showing that for any integer $r$ with $2 \leq r \leq k -2$, we have $\chi(K^2 (2k+r, k)) \leq2 \left(\frac{9}{4}k + \frac{9(r+3)}{8} \right)^r$. Our approach is also related with injective coloring and coloring of Johnson graph.

Abstract:
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 $\chi_l(H)$ be the chromatic number and the list chromatic number of $H$, respectively. A graph $H$ is called {\em chromatic-choosable} if $\chi_l (H) = \chi(H)$. It is an interesting problem to find graphs that are chromatic-choosable. Kostochka and Woodall \cite{KW2001} conjectured that $\chi_l(G^2) = \chi(G^2)$ for every graph $G$, which is called List Square Coloring Conjecture. In this paper, we give infinitely many counterexamples to the conjecture. Moreover, we show that the value $\chi_l(G^2) - \chi(G^2)$ can be arbitrary large.

Abstract:
The Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-elements subsets of an $n$-element set, with two vertices adjacent if the sets are disjoint. 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. Determining the chromatic number of the square of the Kneser graph $K(2k+1, k)$ is an interesting problem, but not much progress has been made. Kim and Nakprasit \cite{2004KN} showed that $\chi(K^2(2k+1,k)) \leq 4k+2$, and Chen, Lih, and Wu \cite{2009CLW} showed that $\chi(K^2(2k+1,k)) \leq 3k+2$ for $k \geq 3$. In this paper, we give improved upper bounds on $\chi(K^2(2k+1,k))$. We show that $\chi(K^2(2k+1,k)) \leq 2k+2$, if $ 2k +1 = 2^n -1$ for some positive integer $n$. Also we show that $\chi(K^2(2k+1,k)) \leq \frac{8}{3}k+\frac{20}{3}$ for every integer $k\ge 2$. In addition to giving improved upper bounds, our proof is concise and can be easily understood by readers while the proof in \cite{2009CLW} is very complicated. Moreover, we show that $\chi(K^2(2k+r,k))=\Theta(k^r)$ for each integer $2 \leq r \leq k-2$.

Abstract:
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.