# Solutions of Some L(2, 1)-Coloring Related Open Problems

Nibedita Mandal; Pratima Panigrahi

Discussiones Mathematicae Graph Theory (2016)

- Volume: 36, Issue: 2, page 279-297
- ISSN: 2083-5892

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topNibedita Mandal, and Pratima Panigrahi. "Solutions of Some L(2, 1)-Coloring Related Open Problems." Discussiones Mathematicae Graph Theory 36.2 (2016): 279-297. <http://eudml.org/doc/277115>.

@article{NibeditaMandal2016,

abstract = {An L(2, 1)-coloring (or labeling) of a graph G is a vertex coloring f : V (G) → Z+ ∪ \{0\} such that |f(u) − f(v)| ≥ 2 for all edges uv of G, and |f(u)−f(v)| ≥ 1 if d(u, v) = 2, where d(u, v) is the distance between vertices u and v in G. The span of an L(2, 1)-coloring is the maximum color (or label) assigned by it. The span of a graph G is the smallest integer λ such that there exists an L(2, 1)-coloring of G with span λ. An L(2, 1)-coloring of a graph with span equal to the span of the graph is called a span coloring. For an L(2, 1)-coloring f of a graph G with span k, an integer h is a hole in f if h ∈ (0, k) and there is no vertex v in G such that f(v) = h. A no-hole coloring is an L(2, 1)-coloring with no hole in it. An L(2, 1)-coloring is irreducible if color of none of the vertices in the graph can be decreased to yield another L(2, 1)-coloring of the same graph. A graph G is inh-colorable if there exists an irreducible no-hole coloring of G. Most of the results obtained in this paper are answers to some problems asked by Laskar et al. [5]. These problems are mainly about relationship between the span and maximum no-hole span of a graph, lower inh-span and upper inh-span of a graph, and the maximum number of holes and minimum number of holes in a span coloring of a graph. We also give some sufficient conditions for a tree and an unicyclic graph to have inh-span Δ + 1.},

author = {Nibedita Mandal, Pratima Panigrahi},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {L(2; 1)-coloring; span of a graph; no-hole coloring; irreducible coloring; unicyclic graph; -coloring},

language = {eng},

number = {2},

pages = {279-297},

title = {Solutions of Some L(2, 1)-Coloring Related Open Problems},

url = {http://eudml.org/doc/277115},

volume = {36},

year = {2016},

}

TY - JOUR

AU - Nibedita Mandal

AU - Pratima Panigrahi

TI - Solutions of Some L(2, 1)-Coloring Related Open Problems

JO - Discussiones Mathematicae Graph Theory

PY - 2016

VL - 36

IS - 2

SP - 279

EP - 297

AB - An L(2, 1)-coloring (or labeling) of a graph G is a vertex coloring f : V (G) → Z+ ∪ {0} such that |f(u) − f(v)| ≥ 2 for all edges uv of G, and |f(u)−f(v)| ≥ 1 if d(u, v) = 2, where d(u, v) is the distance between vertices u and v in G. The span of an L(2, 1)-coloring is the maximum color (or label) assigned by it. The span of a graph G is the smallest integer λ such that there exists an L(2, 1)-coloring of G with span λ. An L(2, 1)-coloring of a graph with span equal to the span of the graph is called a span coloring. For an L(2, 1)-coloring f of a graph G with span k, an integer h is a hole in f if h ∈ (0, k) and there is no vertex v in G such that f(v) = h. A no-hole coloring is an L(2, 1)-coloring with no hole in it. An L(2, 1)-coloring is irreducible if color of none of the vertices in the graph can be decreased to yield another L(2, 1)-coloring of the same graph. A graph G is inh-colorable if there exists an irreducible no-hole coloring of G. Most of the results obtained in this paper are answers to some problems asked by Laskar et al. [5]. These problems are mainly about relationship between the span and maximum no-hole span of a graph, lower inh-span and upper inh-span of a graph, and the maximum number of holes and minimum number of holes in a span coloring of a graph. We also give some sufficient conditions for a tree and an unicyclic graph to have inh-span Δ + 1.

LA - eng

KW - L(2; 1)-coloring; span of a graph; no-hole coloring; irreducible coloring; unicyclic graph; -coloring

UR - http://eudml.org/doc/277115

ER -

## References

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- [7] R.C. Laskar and J.J. Villalpando, Irreducibility of L(2, 1)-coloring and inh-color-ablity of unicyclic and hex graphs, Util. Math. 69 (2006) 65-83. Zbl1123.05040
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