## DIMACS TR: 98-07

## A Characterization for Parity Graphs and a Coloring
Problem with Costs

### Author: Klaus Jansen

**
ABSTRACT
**

In this paper, we give a characterization for parity graphs.
A graph is a parity graph, if and only if for every pair of vertices
all minimal chains joining them have the same parity. We prove
that $G$ is a parity graph, if and only if the cartesian product
$G \times K_2$ is a perfect graph.
Furthermore, as a consequence we get a result for the polyhedron
corresponding to an integer linear program formulation of a
coloring problem with costs. For the case that the costs $k_{v,3} = k_{v,c}$
for each color $c \ge 3$ and vertex $v \in V$, we show that the
polyhedron contains only
integral $0 / 1$ extrema if and only if the graph $G$ is a parity graph.

Paper Available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1998/98-07.ps.gz

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