Tree representations of graphs

  • Authors:

  • Nancy Eaton;Zoltán Füredi;Alexandr V. Kostochka;Jozef Skokan

  • Affiliations:

  • Department of Mathematics, University of Rhode Island, Kingston, RI 02881, USA;Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA and Rényi Institute of Mathematics of the Hungarian Academy of Sciences, Budapest, P. O. Box 127, ...;Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA and Institute of Mathematics, Novosibirsk 630090, Russia;Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA and Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão ...

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2007

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Abstract

A graph is chordal if and only if it is the intersection graph of some family of subtrees of a tree. Applying ''tolerance'' allows larger families of graphs to be represented by subtrees. A graph G is in the family [@D,d,t] if there is a tree with maximum degree @D and subtrees corresponding to the vertices of G such that each subtree has maximum degree at most d and two vertices of G are adjacent if and only if the subtrees corresponding to them have at least t common vertices. It is known that both [3,3,1] and [3,3,2] are equal to the family of chordal graphs. Furthermore, one can easily observe that every graph G belongs to [3,3,t] for some t. Denote by t(G) the minimum t so that G@?[3,3,t]. In this paper, we study t(G) and parameters t(n)=min{t:G@?[3,3,t] for every G@?K"n} and t"b"i"p(n)=min{t:G@?[3,3,t] for every G@?K"n","n}. In particular, our results imply that logn