Bounded degrees and prescribed distances in graphs

Yair Caro, Zsolt Tuza

Research output: Contribution to journalArticle

Abstract

Let X = {x1,...,xm} and Y = {y1,...,ym} be two disjoints sets of vertices in a graph G. Then (X,Y) is called an antipodal set-pair of size m(m-ASP, for short) if the distance of xi and yj is at most two if and only if i ≠ j. We prove that in a graph of maximum degree k every m-ASP has size m≤ k(k + 1) 2 + 1. This upper bound is nearly best possible since, for every k≥2, there exists a regular graph of degree k, with an m-ASP, m≥ k(k+1) 2 (and m = k(k + 1) 2 + 1 when k ≡ 0 or 1 (mod 4)). If the degrees of the xi and yj are bounded above by p and q, respectively, then an m-ASP can exist only for m≤(p + q + 1p). We conjecture that this bound cán be improved to m≤(p + pq), and verify this conjecture when the graph satisfies some additional assumptions.

Original languageEnglish
Pages (from-to)87-93
Number of pages7
JournalDiscrete Mathematics
Volume111
Issue number1-3
DOIs
Publication statusPublished - Feb 22 1993

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ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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