Complexity of equitable tree-coloring problems
A $(q,t)$\emph{-tree-coloring} of a graph $G$ is a $q$-coloring of vertices of $G$ such that the subgraph induced by each color class is a forest of maximum degree at most $t.$ A $(q,\infty)$\emph{-tree-coloring} of a graph $G$ is a $q$-coloring of vertices of $G$ such that the subgraph induced by e...
Saved in:
Main Authors | , |
---|---|
Format | Journal Article |
Language | English |
Published |
30.03.2016
|
Subjects | |
Online Access | Get full text |
Cover
Loading…
Summary: | A $(q,t)$\emph{-tree-coloring} of a graph $G$ is a $q$-coloring of vertices
of $G$ such that the subgraph induced by each color class is a forest of
maximum degree at most $t.$ A $(q,\infty)$\emph{-tree-coloring} of a graph $G$
is a $q$-coloring of vertices of $G$ such that the subgraph induced by each
color class is a forest.
Wu, Zhang, and Li introduced the concept of \emph{equitable $(q,
t)$-tree-coloring} (respectively, \emph{equitable $(q, \infty)$-tree-coloring})
which is a $(q,t)$-tree-coloring (respectively, $(q, \infty)$-tree-coloring)
such that the sizes of any two color classes differ by at most one. Among other
results, they obtained a sharp upper bound on the minimum $p$ such that
$K_{n,n}$ has an equitable $(q, 1)$-tree-coloring for every $q\geq p.$
In this paper, we obtain a polynomial time criterion to decide if a complete
bipartite graph has an equitable $(q,t)$-tree-coloring or an equitable
$(q,\infty)$-tree-coloring. Nevertheless, deciding if a graph $G$ in general
has an equitable $(q,t)$-tree-coloring or an equitable
$(q,\infty)$-tree-coloring is NP-complete. |
---|---|
DOI: | 10.48550/arxiv.1603.09070 |