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# On Chromatic Number of Colored Mixed Graphs

Das, Sandip and Nandi, Soumen and Sen, Sagnik (2017) On Chromatic Number of Colored Mixed Graphs. In: 3rd International Conference on Algorithms and Discrete Applied Mathematics (CALDAM), FEB 16-18, 2017, Sancoale, INDIA, pp. 130-140. PDF Alg_Dis_App_10156-130_2017.pdf - Published Version Restricted to Registered users only Download (145kB) | Request a copy
Official URL: http://dx.doi.org/10.1007/978-3-319-53007-9_12

## 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 x((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 Nektfil and Raspaud (2000, J. Combin. Theory, Ser. B 80, 147-155). They showed that x((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 k(2) k(2+) inverted left perpedicular log((2m+n)) log((2m+n)) k inverted right perpendicular 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 its (m, n)-colored mixed chromatic number is at most 2(Delta-1)(2m+n) (2m + n)(Delta-min(2m+m3)+2) In particular, the last result directly improves the upper bound of 2 Delta(2)2(Delta) oriented chromatic number of graphs with maximum degree Delta, obtained by Kostochka et al. J. Graph Theory 24, 331-340) to 2(Delta-1)(2)2(Delta). We also show that there exists a connected graph with maximum degree Delta and (m, n)-colored mixed chromatic number at least (2m + n)(Delta/2).

Item Type: Conference Paper 3rd International Conference on Algorithms and Discrete Applied Mathematics (CALDAM), Sancoale, INDIA, FEB 16-18, 2017 Copy right for this article belongs to the SPRINGER INTERNATIONAL PUBLISHING AG, GEWERBESTRASSE 11, CHAM, CH-6330, SWITZERLAND Division of Chemical Sciences > Solid State & Structural Chemistry Unit review EPrints Reviewer 12 Jan 2018 06:41 12 Jan 2018 06:41 http://eprints.iisc.ac.in/id/eprint/58765 View Item