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On the chromatic number of (P5,windmill)-free graphs

Schiermeyer I.

Opuscula Mathematica

2017

Vol. 37, no. 4

609--615

In this paper we study the chromatic number of (P5, windmill)-free graphs. For integers r, p ≥ 2 the windmill graph [formula] is the graph obtained by joining a single vertex (the center) to the vertices of p disjoint copies of a complete graph Kr. Our main result is that every (P5, windrnill)-hee graph G admits a polynomial Χ-binding function. Moreover, we will present polynomial Χ-binding functions for several other subclasses of P5-free graphs.

[r, s, t]-colourings of paths

Salvador Villa M., Schiermeyer I.

Opuscula Mathematica

2007

Vol. 27, no. 1

131-149

The concept of [r, s, t]-colourings was recently introduced by Hackmann, Kemnitz and Marangio [3] as follows: Given non-negative integers r, s and t, an [r, s, t]-colouring of a graph G = (V(G), E(G)) is a mapping c from V(G) ∪ E(G) to the colour set {1, 2,..., k} such that ‌c(vi) - c(vj)‌ ≥ r for every two adjacent vertices vi, vj, ‌c(ei) - c(ej)‌ ≥ s for every two adjacent edges ei, ej, and ‌c(vi) - c(ej)‌ ≥ t for all pairs of incident vertices and edges, respectively. The [r, s, t]-chromatic number Xr,s,t(G) of G is defined to be the minimum k such that G admits an [r, s, t]-colouring. In this paper, we determine the [r, s, t]-chromatic number for paths.