33 research outputs found
On coloring parameters of triangle-free planar -graphs
An -graph is a graph with types of arcs and types of edges. A
homomorphism of an -graph to another -graph is a vertex
mapping that preserves the adjacencies along with their types and directions.
The order of a smallest (with respect to the number of vertices) such is
the -chromatic number of .Moreover, an -relative clique of
an -graph is a vertex subset of for which no two distinct
vertices of get identified under any homomorphism of . The
-relative clique number of , denoted by , is the
maximum such that is an -relative clique of . In practice,
-relative cliques are often used for establishing lower bounds of
-chromatic number of graph families.
Generalizing an open problem posed by Sopena [Discrete Mathematics 2016] in
his latest survey on oriented coloring, Chakroborty, Das, Nandi, Roy and Sen
[Discrete Applied Mathematics 2022] conjectured that for any triangle-free planar -graph and that this
bound is tight for all .In this article, we positively settle
this conjecture by improving the previous upper bound of to , and by
finding examples of triangle-free planar graphs that achieve this bound. As a
consequence of the tightness proof, we also establish a new lower bound of for the -chromatic number for the family of triangle-free
planar graphs.Comment: 22 Pages, 5 figure