566 research outputs found
Chordal Graphs are Fully Orientable
Suppose that D is an acyclic orientation of a graph G. An arc of D is called
dependent if its reversal creates a directed cycle. Let m and M denote the
minimum and the maximum of the number of dependent arcs over all acyclic
orientations of G. We call G fully orientable if G has an acyclic orientation
with exactly d dependent arcs for every d satisfying m <= d <= M. A graph G is
called chordal if every cycle in G of length at least four has a chord. We show
that all chordal graphs are fully orientable.Comment: 11 pages, 1 figure, accepted by Ars Combinatoria (March 26, 2010
Full Orientability of the Square of a Cycle
Let D be an acyclic orientation of a simple graph G. An arc of D is called
dependent if its reversal creates a directed cycle. Let d(D) denote the number
of dependent arcs in D. Define m and M to be the minimum and the maximum number
of d(D) over all acyclic orientations D of G. We call G fully orientable if G
has an acyclic orientation with exactly k dependent arcs for every k satisfying
m <= k <= M. In this paper, we prove that the square of a cycle C_n of length n
is fully orientable except n=6.Comment: 7 pages, accepted by Ars Combinatoria on May 26, 201
Induced subgraphs of graphs with large chromatic number. XI. Orientations
Fix an oriented graph H, and let G be a graph with bounded clique number and
very large chromatic number. If we somehow orient its edges, must there be an
induced subdigraph isomorphic to H? Kierstead and Rodl raised this question for
two specific kinds of digraph H: the three-edge path, with the first and last
edges both directed towards the interior; and stars (with many edges directed
out and many directed in). Aboulker et al subsequently conjectured that the
answer is affirmative in both cases. We give affirmative answers to both
questions
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