List injective coloring of planar graphs with girth g≥6

Hong-Yu Chen; Jian-Liang Wu

doi:10.1016/j.disc.2016.06.017

List injective coloring of planar graphs with disjoint 5−-cycles

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921501482 ◽

2021 ◽

pp. 2150148

Author(s):

Wenwen Li ◽

Jiansheng Cai

Keyword(s):

Chromatic Number ◽

Planar Graphs ◽

Maximum Degree ◽

Injective Coloring

An injective [Formula: see text]-coloring of a graph [Formula: see text] is called injective if any two vertices joined by a path of length two get different colors. A graph [Formula: see text] is injectively [Formula: see text]-choosable if for any color list [Formula: see text] of admissible colors on [Formula: see text] of size [Formula: see text] it allows an injective coloring [Formula: see text] such that [Formula: see text] whenever [Formula: see text]. Let [Formula: see text], [Formula: see text] denote the injective chromatic number and injective choosability number of [Formula: see text], respectively. Let [Formula: see text] be a plane with disjoint [Formula: see text]-cycles and maximum degree [Formula: see text]. We show that [Formula: see text] if [Formula: see text], then [Formula: see text]; [Formula: see text] if [Formula: see text], then [Formula: see text].

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LIST INJECTIVE COLORING OF PLANAR GRAPHS WITH GIRTH g ≥ 5

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830914500062 ◽

2014 ◽

Vol 06 (01) ◽

pp. 1450006 ◽

Cited By ~ 2

Author(s):

YUEHUA BU ◽

SHENG YANG

Keyword(s):

Chromatic Number ◽

Planar Graphs ◽

Maximum Degree ◽

Plane Graph ◽

Common Neighbor ◽

Injective Coloring

An injective-k coloring of a graph G is a mapping cV(G) → {1, 2, …, k}, such that c(u) ≠ c(v) for each u, v ∈ V(G), whenever u, v have a common neighbor in G. If G has an injective-k coloring, then we call that G is injective-k colorable. Call χi(G) = min {k | G is injective-k colorable} is the injective chromatic number of G. Assign each vertex v ∈ V(G) a coloring set L(v), then L = {L(v) | v ∈ V(G)} is said to be a color list of G. Let L be a color list of G, if G has an injective coloring c such that c(v) ∈ L(v), ∀v ∈ V(G), then we call c an injective L-coloring of G. If for any color list L, such that |L(v)| ≥ k, G has an injective L-coloring, then G is said to be injective k-choosable. Call [Formula: see text] is injective k-choosable} is the injective chromatic number of G. So far, for the plane graph G of girth g(G) ≥ 5 and maximum degree Δ(G) ≥ 8, the best result of injective chromatic number is χi(G) ≤ Δ + 8. In this paper, for the plane graph G, we proved that [Formula: see text] if girth g(G) ≥ 5 and maximum degree Δ(G) ≥ 8.

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INJECTIVE COLORING OF PLANAR GRAPHS WITH GIRTH 7

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830912500346 ◽

2012 ◽

Vol 04 (02) ◽

pp. 1250034 ◽

Cited By ~ 6

Author(s):

YUEHUA BU ◽

KAI LU

Keyword(s):

Planar Graph ◽

Chromatic Number ◽

Planar Graphs ◽

Common Neighbor ◽

Injective Coloring

An injective k-coloring of a graph G is an assignment of k colors to the vertices of G so that any two vertices with a common neighbor receive distinct colors, and χi(G) is the injective chromatic number of G. Dimitrov et al. proved χi(G) ≤ Δ(G) + 2 for a planar graph G with g(G) ≥ 7. In this paper, we show that if G is a planar graph with g(G) ≥ 7 and Δ(G) ≥ 7, then χi(G) ≤ Δ(G) + 1.

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Injective coloring of planar graphs

Discrete Applied Mathematics ◽

10.1016/j.dam.2008.08.016 ◽

2009 ◽

Vol 157 (4) ◽

pp. 663-672 ◽

Cited By ~ 25

Author(s):

Yuehua Bu ◽

Dong Chen ◽

André Raspaud ◽

Weifan Wang

Keyword(s):

Planar Graphs ◽

Injective Coloring

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List injective coloring of planar graphs with girth 5, 6, 8

Discrete Applied Mathematics ◽

10.1016/j.dam.2012.12.017 ◽

2013 ◽

Vol 161 (10-11) ◽

pp. 1367-1377 ◽

Cited By ~ 4

Author(s):

Yuehua Bu ◽

Kai Lu

Keyword(s):

Planar Graphs ◽

Injective Coloring

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List injective coloring of a class of planar graphs without short cycles

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830918500684 ◽

2018 ◽

Vol 10 (05) ◽

pp. 1850068

Author(s):

Yuehua Bu ◽

Chaoyuan Huang

Keyword(s):

Planar Graph ◽

Planar Graphs ◽

Common Neighbor ◽

Injective Coloring

An injective [Formula: see text]-coloring of a graph [Formula: see text] is a mapping c: [Formula: see text]([Formula: see text]) [Formula: see text][Formula: see text] such that [Formula: see text] whenever [Formula: see text] have a common neighbor in [Formula: see text]. A list assignment of a graph [Formula: see text] is a mapping [Formula: see text] that assigns a color list [Formula: see text] to each vertex [Formula: see text]. Given a list assignment [Formula: see text] of [Formula: see text], an injective coloring [Formula: see text] of [Formula: see text] is called an injective [Formula: see text]-coloring if [Formula: see text] for every [Formula: see text]. In this paper, we show that if [Formula: see text] is a planar graph with girth [Formula: see text], then [Formula: see text] if [Formula: see text].

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