5 research outputs found
Competitively tight graphs
The competition graph of a digraph is a (simple undirected) graph which
has the same vertex set as and has an edge between two distinct vertices
and if and only if there exists a vertex in such that
and are arcs of . For any graph , together with sufficiently
many isolated vertices is the competition graph of some acyclic digraph. The
competition number of a graph 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 is related to the edge clique cover number of the
graph via . We first show
that for any positive integer satisfying , there
exists a graph with and characterize a graph
satisfying . We then focus on what we call
\emph{competitively tight graphs} which satisfy the lower bound, i.e.,
. 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
Boxicity of graphs on surfaces
The boxicity of a graph is the least integer for which there
exist interval graphs , , such that . 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 of
genus is at most . This result yields improved bounds on the
dimension of the adjacency poset of graphs on surfaces.Comment: 9 pages, 2 figure