Contractions and minimal k-colorability

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6 Citations (Scopus)

Abstract

Coloring the vertex set of a graph G with positive integers, the chromatic sum Σ(G) of G is the minimum sum of colors in a proper coloring. The strength of G is the largest integer that occurs in every coloring whose total is Σ(G). Proving a conjecture of Kubicka and Schwenk, we show that every tree of strength s has at least ((2 + {Mathematical expression})s-1 - (2 - {Mathematical expression})s-1)/ {Mathematical expression} vertices (s ≥ 2). Surprisingly, this extremal result follows from a topological property of trees. Namely, for every s ≥ 3 there exist precisely two trees Ts and Rs such that every tree of strength at least s is edge-contractible to Ts or Rs.

Original languageEnglish
Pages (from-to)51-59
Number of pages9
JournalGraphs and Combinatorics
Volume6
Issue number1
DOIs
Publication statusPublished - Mar 1 1990

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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