Packing k-edge trees in graphs of restricted vertex degrees
Let ${\cal G}^{s}_{r}$ denote the set of graphs with each vertex of degree at least r and at most s, v(G) the number of vertices, and τk (G) the maximum number of disjoint k‐edge trees in G. In this paper we show that (a1) if G ∈ ${\cal G}^{s}_{2}$ and s ≥ 4, then τ2(G) ≥ v(G)/(s + 1), (a2) if G ∈ $...
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| Published in: | Journal of graph theory Vol. 55; no. 4; pp. 306 - 324 |
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| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Hoboken
Wiley Subscription Services, Inc., A Wiley Company
01.08.2007
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| Subjects: | |
| ISSN: | 0364-9024, 1097-0118 |
| Online Access: | Get full text |
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| Summary: | Let ${\cal G}^{s}_{r}$ denote the set of graphs with each vertex of degree at least r and at most s, v(G) the number of vertices, and τk (G) the maximum number of disjoint k‐edge trees in G. In this paper we show that
(a1)
if G ∈ ${\cal G}^{s}_{2}$ and s ≥ 4, then τ2(G) ≥ v(G)/(s + 1),
(a2)
if G ∈ ${\cal G}^{3}_{2}$ and G has no 5‐vertex components, then τ2(G) ≥ v(G)4,
(a3)
if G ∈ ${\cal G}^{s}_{1}$ and G has no k‐vertex component, where k ≥ 2 and s ≥ 3, then τk(G) ≥ (v(G) ‐k)/(sk ‐ k + 1), and
(a4)
the above bounds are attained for infinitely many connected graphs.
Our proofs provide polynomial time algorithms for finding the corresponding packings in a graph. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 306–324, 2007 |
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| Bibliography: | istex:BF0F65410DAE2347C47538C0768888351A291985 ark:/67375/WNG-794PBLG6-F ArticleID:JGT20238 |
| ISSN: | 0364-9024 1097-0118 |
| DOI: | 10.1002/jgt.20238 |