[1,1,t]-Colorings of Complete Graphs

Arnfried Kemnitz, Massimiliano Marangio, Zsolt Tuza

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

Given non-negative integers r, s, and t, an [r,s,t]-coloring of a graph G = (V(G),E(G)) is a mapping c from V(G) ∪ E(G) to the color set {1,...,k} such that {pipe}c(vi) - c(vj){pipe} ≥ r for every two adjacent vertices vi,vj, {pipe}c(ei) - c(ej){pipe} ≥ s for every two adjacent edges ei,ej, and {pipe}c(vi) - c(ej){pipe} ≥ t for all pairs of incident vertices and edges, respectively. The [r,s,t]-chromatic number Χr,s,t(G) of G is defined to be the minimum k such that G admits an [r,s,t]-coloring. In this note we examine χ 1,1,t(Kp) for complete graphs Kp. We prove, among others, that χ 1,1,t(Kp) is equal to p+t-2+min{p,t} whenever t ≥ [p/2]-1, but is strictly larger if p is even and sufficiently large with respect to t. Moreover, as p → ∞ and t=t(p), we asymptotically have χ 1,1,t(Kp)=p+o(p) if and only if t=o(p).

Original languageEnglish
Pages (from-to)1041-1050
Number of pages10
JournalGraphs and Combinatorics
Volume29
Issue number4
DOIs
Publication statusPublished - Jul 1 2013

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Keywords

  • Complete graphs
  • Generalized colorings
  • [r, s, t]-Chromatic number
  • [r, s, t]-Colorings

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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