74 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

### A Note on Near-factor-critical Graphs

A near-factor of a finite simple graph $G$ is a matching that saturates all
vertices except one. A graph $G$ is said to be near-factor-critical if the
deletion of any vertex from $G$ results in a subgraph that has a near-factor.
We prove that a connected graph $G$ is near-factor-critical if and only if it
has a perfect matching. We also characterize disconnected near-factor-critical
graphs.Comment: 4 page

### 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

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