# On chromatic number of colored mixed graphs

## Document Type

Conference Article

## Publication Title

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

## Abstract

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 an (m, n)- colored mixed graph G is the order (number of vertices) of a smallest homomorphic image of G. This notion was introduced by Nešetřil and Raspaud (2000, J. Combin. Theory, Ser. B 80, 147–155). They showed that χ(m, n)(G) ≤ k(2m + n)k−1 where G is a acyclic k-colorable graph. We prove the tightness of this bound. We also show that the acyclic chromatic number of a graph is bounded by k2 + k2+ [log(2m+n) log(2m+n) k] if its (m, n)-colored mixed chromatic number is at most k. Furthermore, using probabilistic method, we show that for connected graphs with maximum degree Δ its (m, n)-colored mixed chromatic number is at most 2(Δ − 1)2m+n(2m + n)Δ−min(2m+n,3)+2. In particular, the last result directly improves the upper bound 2Δ22Δ of oriented chromatic number of graphs with maximum degree Δ, obtained by Kostochka et al. (1997, J. Graph Theory 24, 331–340) to 2(Δ− 1)22Δ. We also show that there exists a connected graph with maximum degree Δ and (m, n)-colored mixed chromatic number at least (2m + n)Δ/2.

## First Page

130

## Last Page

140

## DOI

10.1007/978-3-319-53007-9_12

## Publication Date

1-1-2017

## Recommended Citation

Das, Sandip; Nandi, Soumen; and Sen, Sagnik, "On chromatic number of colored mixed graphs" (2017). *Conference Articles*. 349.

https://digitalcommons.isical.ac.in/conf-articles/349

## Comments

Open Access, Green