Adjacent vertex-distinguishing edge coloring of graphs with maximum degree Δ

The adjacent vertex distinguishing edge coloring of a graph [Formula: see text] is a proper edge coloring of [Formula: see text] such that the color sets of any pair of adjacent vertices are distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of [Formula: see text] is denoted by [Formula: see text]. It is observed that [Formula: see text] when [Formula: see text] contains two adjacent vertices of degree [Formula: see text]. In this paper, we prove that if [Formula: see text] is a planar graph without 3-cycles, then [Formula: see text]. Furthermore, we characterize the adjacent vertex distinguishing chromatic index for planar graphs of [Formula: see text] and without 3-cycles. This improves a result from [D. Huang, Z. Miao and W. Wang, Adjacent vertex distinguishing indices of planar graphs without 3-cycles, Discrete Math. 338 (2015) 139–148] that established [Formula: see text] for planar graphs without 3-cycles.

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Adjacent vertex distinguishing total coloring of planar graphs with maximum degree 9

Discrete Mathematics ◽

10.1016/j.disc.2019.01.024 ◽

2019 ◽

Vol 342 (5) ◽

pp. 1392-1402

Author(s):

Jie Hu ◽

Guanghui Wang ◽

Jianliang Wu ◽

Donglei Yang ◽

Xiaowei Yu

Keyword(s):

Planar Graphs ◽

Maximum Degree ◽

Total Coloring ◽

Adjacent Vertex

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List strong edge coloring of planar graphs with maximum degree 4

Discrete Mathematics ◽

10.1016/j.disc.2018.10.034 ◽

2019 ◽

Vol 342 (5) ◽

pp. 1471-1480

Author(s):

Ming Chen ◽

Jie Hu ◽

Xiaowei Yu ◽

Shan Zhou

Keyword(s):

Planar Graphs ◽

Maximum Degree ◽

Edge Coloring

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Adjacent vertex distinguishing total coloring of planar graphs with large maximum degree

Scientia Sinica Mathematica ◽

10.1360/012011-359 ◽

2012 ◽

Vol 42 (2) ◽

pp. 151-164 ◽

Cited By ~ 24

Author(s):

WeiFan WANG ◽

DanJun HUANG

Keyword(s):

Planar Graphs ◽

Maximum Degree ◽

Total Coloring ◽

Adjacent Vertex ◽

Large Maximum

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On the Adjacent Vertex Distinguishing Proper Edge Colorings of Several Classes of Complete 5-Partite Graphs

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.333-335.1452 ◽

2013 ◽

Vol 333-335 ◽

pp. 1452-1455

Author(s):

Chun Yan Ma ◽

Xiang En Chen ◽

Fang Yang ◽

Bing Yao

Keyword(s):

Chromatic Number ◽

Edge Coloring ◽

Adjacent Vertex ◽

Chromatic Numbers ◽

Edge Colorings ◽

Proper Edge Coloring ◽

Minimum Number

A proper $k$-edge coloring of a graph $G$ is an assignment of $k$ colors, $1,2,\cdots,k$, to edges of $G$. For a proper edge coloring $f$ of $G$ and any vertex $x$ of $G$, we use $S(x)$ denote the set of thecolors assigned to the edges incident to $x$. If for any two adjacent vertices $u$ and $v$ of $G$, we have $S(u)\neq S(v)$,then $f$ is called the adjacent vertex distinguishing proper edge coloring of $G$ (or AVDPEC of $G$ in brief). The minimum number of colors required in an AVDPEC of $G$ is called the adjacent vertex distinguishing proper edge chromatic number of $G$, denoted by $\chi^{'}_{\mathrm{a}}(G)$. In this paper, adjacent vertex distinguishing proper edge chromatic numbers of several classes of complete 5-partite graphs are obtained.

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