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Backbone coloring for triangle-free planar graphs

Backbone coloring for triangle-free planar graphs Let G be a graph and H a subgraph of G. A backbone-k-coloring of (G, H) is a mapping f: V(G) → {1, 2, ···, k} such that |f(u) − f(v)| ≥ 2 if uv ∈ E(H) and |f(u) − f(v)| ≥ 1 if uv ∈ E(G)E(H). The backbone chromatic number of (G, H) denoted by χ b (G, H) is the smallest integer k such that (G, H) has a backbone-k-coloring. In this paper, we prove that if G is either a connected triangle-free planar graph or a connected graph with mad(G) < 3, then there exists a spanning tree T of G such that χ b (G, T) ≤ 4. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Backbone coloring for triangle-free planar graphs

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References (10)

Publisher
Springer Journals
Copyright
Copyright © 2017 by Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag GmbH Germany
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-017-0700-3
Publisher site
See Article on Publisher Site

Abstract

Let G be a graph and H a subgraph of G. A backbone-k-coloring of (G, H) is a mapping f: V(G) → {1, 2, ···, k} such that |f(u) − f(v)| ≥ 2 if uv ∈ E(H) and |f(u) − f(v)| ≥ 1 if uv ∈ E(G)E(H). The backbone chromatic number of (G, H) denoted by χ b (G, H) is the smallest integer k such that (G, H) has a backbone-k-coloring. In this paper, we prove that if G is either a connected triangle-free planar graph or a connected graph with mad(G) < 3, then there exists a spanning tree T of G such that χ b (G, T) ≤ 4.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Aug 7, 2017

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