Chromatic number of sparse colored mixed planar graphs

Chromatic number of sparse colored mixed planar graphs

A colored mixed graph has vertices linked by both colored arcs and colored edges. The chromatic number of such a graph G is defined as the smallest order of a colored mixed graph H such that there exists a (arc-color preserving) homomorphism from G to H . We study in this paper the colored mixed chromatic number of planar graphs, partial 2-trees and outerplanar graphs with given girth.

On Chromatic Number of Colored Mixed Graphs

An (m,n)-colored mixed graph G is a graph with its arcs having one of the m different colors and edges having one of the n different colors. A homomorphism f of an (m,n)colored mixed graph G to an (m,n)-colored mixed graph H is a vertex mapping such that if uv is an arc (edge) of color c in G, then f(u)f(v) is an arc (edge) of color c in H . The (m,n)-colored mixed chromatic number χ(m,n)(G) of...

The Chromatic Number of Sparse Graphs

The locating-chromatic number for Halin graphs

Let G be a connected graph. Let f be a proper k -coloring of G and Π = (R_1, R_2, . . . , R_k) bean ordered partition of V (G) into color classes. For any vertex v of G, define the color code c_Π(v) of v with respect to Π to be a k -tuple (d(v, R_1), d(v, R_2), . . . , d(v, R_k)), where d(v, R_i) is the min{d(v, x)|x ∈ R_i}. If distinct vertices have distinct color codes, then we call f a locat...

Precise upper bound for the strong edge chromatic number of sparse planar graphs

We prove that every planar graph with maximum degree ∆ is strong edge (2∆− 1)-colorable if its girth is at least 40⌊ 2 ⌋+1. The bound 2∆− 1 is reached at any graph that has two adjacent vertices of degree ∆.