Extremal jumps of the Hall number

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Abstract

Given a graph G = (V, E) and sets L (v) of allowed colors for each v ∈ V, a list coloring of G is an assignment of colors φ (v) to the vertices, such that φ (v) ∈ L (v) for all v ∈ V and φ (u) ≠ φ (v) for all u v ∈ E. The Hall number of G is the smallest positive integer k such that G admits a list coloring provided that | L (v) | ≥ k for every vertex v and, for every X ⊆ V, the sum-over all colors c-of the maximum number of independent vertices in X whose lists contain c, is at least | X |. We prove that every graph of order n ≥ 3 has Hall number at most n - 2. Combining this upper bound with a construction, we deduce that vertex deletion or edge insertion in a graph of order n > 3 may make the Hall number decrease by as much as n - 3, and that this estimate is tight for all n. This solves two problems raised by Hilton, Johnson and Wantland [Discrete Applied Math. 94 (1999), 227-245].

Original languageEnglish
Pages (from-to)83-89
Number of pages7
JournalElectronic Notes in Discrete Mathematics
Volume28
DOIs
Publication statusPublished - Mar 1 2007

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Keywords

  • Graph
  • Hall condition
  • Hall number
  • list coloring
  • vertex coloring

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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