id: 06046354 dt: j an: 06046354 au: Kim, Suh-Ryung; Lee, Jung Yeun; Park, Boram; Sano, Yoshio ti: The competition number of a graph and the dimension of its hole space. so: Appl. Math. Lett. 25, No. 3, 638-642 (2012). py: 2012 pu: Elsevier Science Ltd. (Pergamon), Oxford la: EN cc: ut: competition graph; competition number; cycle space; hole; hole space ci: li: doi:10.1016/j.aml.2011.10.003 ab: Summary: 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. rv: