Crossing-critical graphs with large maximum degree

Suppose that G is a graph with maximum degree d(G) such that, for every vertex v in G, the neighbourhood of v contains at most d(G)2/f (f > 1) edges. We show that the list chromatic number of G is at most Kd(G)/log f, for some positive constant K. This result is sharp up to the multiplicative constant K and strengthens previous results by Kim [9], Johansson [7], Alon, Krivelevich and Sudakov [3], and the present author [18]. This also motivates several interesting questions.As an application, we derive several upper bounds for the strong (list) chromatic index of a graph, under various assumptions. These bounds extend earlier results by Faudree, Gyárfás, Schelp and Tuza [6] and Mahdian [13] and determine, up to a constant factor, the strong (list) chromatic index of a random graph. Another application is an extension of a result of Kostochka and Steibitz [10] concerning the structure of list critical graphs.

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REMARKS ON EDGE CRITICAL GRAPHS WITH MAXIMUM DEGREE OF 3 AND 4

Advances and Applications in Discrete Mathematics ◽

10.17654/dm018040505 ◽

2017 ◽

Vol 18 (4) ◽

pp. 505-514

Author(s):

Suechao Li ◽

Xuechao Li ◽

Bing Wei

Keyword(s):

Maximum Degree ◽

Critical Graphs

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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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Fractionally Edge Colouring Graphs with Large Maximum Degree in Linear Time

Electronic Notes in Discrete Mathematics ◽

10.1016/j.endm.2009.07.008 ◽

2009 ◽

Vol 34 ◽

pp. 47-51

Author(s):

W. Sean Kennedy ◽

Conor Meagher ◽

Bruce A. Reed

Keyword(s):

Maximum Degree ◽

Linear Time ◽

Large Maximum ◽

Edge Colouring

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On the average degree of critical graphs with maximum degree six

Discrete Mathematics ◽

10.1016/j.disc.2011.06.029 ◽

2011 ◽

Vol 311 (21) ◽

pp. 2574-2576 ◽

Cited By ~ 2

Author(s):

Lianying Miao ◽

Jibin Qu ◽

Qingbo Sun

Keyword(s):

Maximum Degree ◽

Average Degree ◽

Critical Graphs

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The classification of f -coloring of graphs with large maximum degree

Applied Mathematics and Computation ◽

10.1016/j.amc.2017.05.059 ◽

2017 ◽

Vol 313 ◽

pp. 119-121 ◽

Cited By ~ 1

Author(s):

Jiansheng Cai ◽

Guiying Yan ◽

Xia Zhang

Keyword(s):

Maximum Degree ◽

Large Maximum

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The Distance-t Chromatic Index of Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548313000473 ◽

2013 ◽

Vol 23 (1) ◽

pp. 90-101 ◽

Cited By ~ 7

Author(s):

TOMÁŠ KAISER ◽

ROSS J. KANG

Keyword(s):

Maximum Degree ◽

Upper Bounds ◽

The Other ◽

Graph Colouring ◽

Chromatic Index ◽

Multiplicative Factor ◽

Strong Chromatic Index ◽

Absolute Positive Constant ◽

Large Maximum ◽

Fixed Positive Integer

We consider two graph colouring problems in which edges at distance at most t are given distinct colours, for some fixed positive integer t. We obtain two upper bounds for the distance-t chromatic index, the least number of colours necessary for such a colouring. One is a bound of (2-ε)Δt for graphs of maximum degree at most Δ, where ε is some absolute positive constant independent of t. The other is a bound of O(Δt/log Δ) (as Δ → ∞) for graphs of maximum degree at most Δ and girth at least 2t+1. The first bound is an analogue of Molloy and Reed's bound on the strong chromatic index. The second bound is tight up to a constant multiplicative factor, as certified by a class of graphs of girth at least g, for every fixed g ≥ 3, of arbitrarily large maximum degree Δ, with distance-t chromatic index at least Ω(Δt/log Δ).

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Peg solitaire on graphs with large maximum degree

Asian-European Journal of Mathematics ◽

10.1142/s1793557122500577 ◽

2021 ◽

pp. 2250057

Author(s):

Naoki Matsumoto

Keyword(s):

Connected Graph ◽

Maximum Degree ◽

Terminal State ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Combinatorial Game ◽

Starting State ◽

Large Maximum ◽

Necessary And Sufficient

In 2011, Beeler and Hoilman introduced the peg solitaire on graphs. The peg solitaire on a connected graph is a one-player combinatorial game starting with exactly one hole in a vertex and pegs in all other vertices and removing all pegs but exactly one by a sequence of jumps; for a path [Formula: see text], if there are pegs in [Formula: see text] and [Formula: see text] and exists a hole in [Formula: see text], then [Formula: see text] can jump over [Formula: see text] into [Formula: see text], and after that, the peg in [Formula: see text] is removed. A problem of interest in the game is to characterize solvable (respectively, freely solvable) graphs, where a graph is solvable (respectively, freely solvable) if for some (respectively, any) vertex [Formula: see text], starting with a hole [Formula: see text], a terminal state consisting of a single peg can be obtained from the starting state by a sequence of jumps. In this paper, we consider the peg solitaire on graphs with large maximum degree. In particular, we show the necessary and sufficient condition for a graph with large maximum degree to be solvable in terms of the number of pendant vertices adjacent to a vertex of maximum degree. It is a notable point that this paper deals with a question of Beeler and Walvoort whether a non-solvable condition of trees can be extended to other graphs.

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