On packing 3-vertex paths in a graph

Atsushi Kaneko, Alexander Kelmans, Tsuyoshi Nishimura

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15 Citations (Scopus)

Abstract

Let G be a connected graph and let eb(G) and λ(G) denote the number of end-blocks and the maximum number of disjoint 3-vertex paths Λ in G. We prove the following theorems on claw-free graphs: (t1) if G is claw-free and eb(G) ≤ 2 (and in particular, G is 2-connected) then λ(G) = ⌊|V(G)|/3⌋; (t2) if G is claw-free and eb(G) ≥ 2 then λ(G) ≥ ⌊(|V(G)|-eb(G) + 2)/3⌋; and (t3) if G is claw-free and Δ*-free then λ(G) = ⌊|V(G)|/3⌋ (here Δ* is a graph obtained from a triangle Δ by attaching to each vertex a new dangling edge). We also give the following sufficient condition for a graph to have a Λ-factor: Let n and p be integers, 1 ≤ p ≤ n - 2, G a 2-connected graph, and |V(G)| =3n. Suppose that G-S has a Λ-factor for every S ⊆ V(G) such that |S| = 3p and both V(G)-S and S induce connected subgraphs in G. Then G has a Λ-factor.

Original languageEnglish
Pages (from-to)175-197
Number of pages23
JournalJournal of Graph Theory
Volume36
Issue number4
Publication statusPublished - 2001 Apr

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Keywords

  • Claw-free graphs
  • Path factors
  • Path packings

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Kaneko, A., Kelmans, A., & Nishimura, T. (2001). On packing 3-vertex paths in a graph. Journal of Graph Theory, 36(4), 175-197.