Discussiones Mathematicae Graph Theory 30(3) (2010) 449-459
doi: 10.7151/dmgt.1506

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THE COMPETITION NUMBERS OF JOHNSON GRAPHS

Suh-Ryung Kim,  Boram Park

Department of Mathematics Education
Seoul National University, Seoul 151-742, Korea

Yoshio Sano

Research Institute for Mathematical Sciences
Kyoto University, Kyoto 606-8502, Japan

Abstract

The competition graph of a digraph D is a 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 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 to characterize all graphs with given competition number k has been one of the important research problems in the study of competition graphs.

The Johnson graph J(n,d) has the vertex set {vX | X ∈ ([n] || d)}, where ([n] || d) denotes the set of all d-subsets of an n-set [n] = {1, …, n }, and two vertices vX1 and vX2 are adjacent if and only if |X1 ∩X2| = d−1. In this paper, we study the edge clique number and the competition number of J(n,d). Especially we give the exact competition numbers of J(n,2) and J(n,3).

Keywords: competition graph, competition number, edge clique cover, Johnson graph.

2010 Mathematics Subject Classification: 05C69, 05C75.

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Received 14 April 2009
Revised 9 October 2009
Accepted 10 October 2009