Improved upper bound for acyclic chromatic index of planar graphs without 4-cycles

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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An improved upper bound for the acyclic chromatic number of 1-planar graphs

Discrete Applied Mathematics ◽

10.1016/j.dam.2020.01.010 ◽

2020 ◽

Vol 283 ◽

pp. 275-291

Author(s):

Wanshun Yang ◽

Weifan Wang ◽

Yiqiao Wang

Keyword(s):

Chromatic Number ◽

Upper Bound ◽

Planar Graphs

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Precise upper bound for the strong edge chromatic number of sparse planar graphs

Discussiones Mathematicae Graph Theory ◽

10.7151/dmgt.1708 ◽

2013 ◽

Vol 33 (4) ◽

pp. 756 ◽

Cited By ~ 11

Author(s):

Oleg V. Borodin ◽

Anna O. Ivanova

Keyword(s):

Chromatic Number ◽

Upper Bound ◽

Planar Graphs

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Counting Triangulations of Planar Point Sets

The Electronic Journal of Combinatorics ◽

10.37236/557 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 25

Author(s):

Micha Sharir ◽

Adam Sheffer

Keyword(s):

Upper Bound ◽

Planar Graphs ◽

Spanning Trees ◽

Upper Bounds ◽

Line Graphs ◽

Point Sets ◽

Planar Point ◽

Straight Line ◽

Point Set ◽

Spanning Cycles

We study the maximal number of triangulations that a planar set of $n$ points can have, and show that it is at most $30^n$. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl (2006), which has led to the previous best upper bound of $43^n$ for the problem. Moreover, this new bound is useful for bounding the number of other types of planar (i.e., crossing-free) straight-line graphs on a given point set. Specifically, it can be used to derive new upper bounds for the number of planar graphs ($207.84^n$), spanning cycles ($O(68.67^n)$), spanning trees ($O(146.69^n)$), and cycle-free graphs ($O(164.17^n)$).

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