Jung Yeun Lee

The competition number of a graph whose holes do not overlap much

Let D be an acyclic digraph. The competition graph of D is a graph which has the same vertex set as D and has an edge between u and v if and only if there exists a vertex x in D such that (u,x) and (v,x) 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. A hole of a graph is an induced cycle of length at least four. Kim (2005) [8] conjectured that the competition number of a graph with h holes is at most h+1. Recently, Li and Chang (2009) [11] showed that the conjecture is true when the holes are independent. In this paper, we show that the conjecture is true though the holes are not independent but mutually edge-disjoint.

The m-step competition graphs of doubly partial orders

Abstract The competition graph of a doubly partial order is an interval graph. The competition-common enemy graph, a variant of the competition graph, of a doubly partial order is also an interval graph if it does not contain a 4-cycle as an induced subgraph. It is natural to ask whether or not the same phenomenon occurs for other interesting variants of the competition graph. In this paper, we study the m -step competition graph, a generalization of the competition graph, of a doubly partial order. We show that the m -step competition graph of a doubly partial order is an interval graph for every positive integer m . We also show that given a positive integer m , an interval graph with sufficiently many isolated vertices is the m -step competition graph of a doubly partial order.

The competition hypergraphs of doubly partial orders

Since Cho and Kim (2005) [2] showed that the competition graph of a doubly partial order is an interval graph, it has been actively studied whether or not the same phenomenon occurs for other variants of competition graphs and interesting results have been obtained. Continuing in the same spirit, we study the competition hypergraph, an interesting variant of the competition graph, of a doubly partial order. Though it turns out that the competition hypergraph of a doubly partial order is not always interval, we completely characterize the competition hypergraphs of doubly partial orders which are interval.

Graphs having many holes but with small competition numbers

Abstract The competition number k ( G ) of a graph G is the smallest number k such that G together with k isolated vertices added is the competition graph of an acyclic digraph. A chordless cycle of length at least 4 of a graph is called a hole of the graph. The number of holes of a graph is closely related to its competition number as the competition number of a graph which does not contain a hole is at most one and the competition number of a complete bipartite graph K ⌊ n 2 ⌋ , ⌈ n 2 ⌉ which has so many holes that no more holes can be added is the largest among those of graphs with n vertices. In this paper, we show that even if a connected graph G has many holes, the competition number of G can be as small as 2 under some assumption. In addition, we show that, for a connected graph G with exactly h holes and at most one non-edge maximal clique, if all the holes of G are pairwise edge-disjoint and the clique number ω = ω ( G ) of G satisfies 2 ≤ ω ≤ h + 1 , then the competition number of G is at most h − ω + 3 .

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

Abstract 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 the 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 has been 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 not 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.

On the competition graphs of d -partial orders

In this paper, we study the competition graphs of

A generalization of Opsut's result on the competition numbers of line graphs

d

Competitively Tight Graphs

-partial orders and obtain their characterization which extends results given by Cho and Kim \cite{chokim} in 2005. We also show that any graph can be made into the competition graph of a

On the partial order competition dimensions of chordal graphs

d

The competition graphs of oriented complete bipartite graphs

-partial order for some positive integer