The competition numbers of ternary Hamming graphs

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.Comment: 6 pages, 2 figure

Competitively tight graphs

The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between two distinct vertices $x$ 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 the smallest number of such isolated vertices. Computing the competition number of a graph is an NP-hard problem in general and has been one of the important research problems in the study of competition graphs. Opsut [1982] showed that the competition number of a graph $G$ is related to the edge clique cover number $\theta_E(G)$ of the graph $G$ via $\theta_E(G)-|V(G)|+2 \leq k(G) \leq \theta_E(G)$. We first show that for any positive integer $m$ satisfying $2 \leq m \leq |V(G)|$, there exists a graph $G$ with $k(G)=\theta_E(G)-|V(G)|+m$ and characterize a graph $G$ satisfying $k(G)=\theta_E(G)$. We then focus on what we call \emph{competitively tight graphs} $G$ which satisfy the lower bound, i.e., $k(G)=\theta_E(G)-|V(G)|+2$. We completely characterize the competitively tight graphs having at most two triangles. In addition, we provide a new upper bound for the competition number of a graph from which we derive a sufficient condition and a necessary condition for a graph to be competitively tight.Comment: 10 pages, 2 figure

The competition number of a graph and the dimension of its hole space

The competition graph of a digraph D is a (simple undirected) 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 G is 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 to characterize a graph by its competition number. Recently, the relationship between the competition number and the number of holes of a graph is being studied. A hole of a graph is a cycle of length at least 4 as an induced subgraph. In this paper, we conjecture that the dimension of the hole space of a graph is no smaller than the competition number of the graph. We verify this conjecture for various kinds of graphs and show that our conjectured inequality is indeed an equality for connected triangle-free graphs.Comment: 6 pages, 3 figure

Boxicity of graphs on surfaces

The boxicity of a graph $G=(V,E)$ is the least integer $k$ for which there exist $k$ interval graphs $G_i=(V,E_i)$, $1 \le i \le k$, such that $E=E_1 \cap ... \cap E_k$. Scheinerman proved in 1984 that outerplanar graphs have boxicity at most two and Thomassen proved in 1986 that planar graphs have boxicity at most three. In this note we prove that the boxicity of toroidal graphs is at most 7, and that the boxicity of graphs embeddable in a surface $\Sigma$ of genus $g$ is at most $5g+3$. This result yields improved bounds on the dimension of the adjacency poset of graphs on surfaces.Comment: 9 pages, 2 figure

Optimal k-fold colorings of webs and antiwebs

A k-fold x-coloring of a graph is an assignment of (at least) k distinct colors from the set {1, 2, ..., x} to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The smallest number x such that G admits a k-fold x-coloring is the k-th chromatic number of G, denoted by \chi_k(G). We determine the exact value of this parameter when G is a web or an antiweb. Our results generalize the known corresponding results for odd cycles and imply necessary and sufficient conditions under which \chi_k(G) attains its lower and upper bounds based on the clique, the fractional chromatic and the chromatic numbers. Additionally, we extend the concept of \chi-critical graphs to \chi_k-critical graphs. We identify the webs and antiwebs having this property, for every integer k <= 1.Comment: A short version of this paper was presented at the Simp\'osio Brasileiro de Pesquisa Operacional, Brazil, 201