4 research outputs found
Coloring Grids Avoiding Bicolored Paths
The vertex-coloring problem on graphs avoiding bicolored members of a family
of subgraphs has been widely studied. Most well-known examples are star
coloring and acyclic coloring of graphs (Gr\"unbaum, 1973) where bicolored
copies of and cycles are not allowed, respectively. In this paper, we
study a variation of this problem, by considering vertex coloring on grids
forbidding bicolored paths. We let -chromatic number of a graph be the
minimum number of colors needed to color the vertex set properly avoiding a
bicolored We show that in any 3-coloring of the cartesian product of
paths, , there is a bicolored With our result,
the problem of finding the -chromatic number of product of two paths
(2-dimensional grid) is settled for all $k.
Coloring of Graphs Avoiding Bicolored Paths of a Fixed Length
The problem of finding the minimum number of colors to color a graph properly
without containing any bicolored copy of a fixed family of subgraphs has been
widely studied. Most well-known examples are star coloring and acyclic coloring
of graphs (Gr\"unbaum, 1973) where bicolored copies of and cycles are not
allowed, respectively. In this paper, we introduce a variation of these
problems and study proper coloring of graphs not containing a bicolored path of
a fixed length and provide general bounds for all graphs. A -coloring of
an undirected graph is a proper vertex coloring of such that there is
no bicolored copy of in and the minimum number of colors needed for
a -coloring of is called the -chromatic number of denoted by
We provide bounds on for all graphs, in particular, proving
that for any graph with maximum degree and
Moreover, we find the exact values for the
-chromatic number of the products of some cycles and paths for $k=5,6.