A note on the strong edge-coloring of outerplanar graphs with maximum degree 3

2016 ◽  
Vol 32 (4) ◽  
pp. 883-890
Author(s):  
Shun-yi Liu ◽  
He-ping Zhang ◽  
Hong-liang Lu ◽  
Yu-qing Lin
Keyword(s):  
Maximum Degree ◽  
Edge Coloring ◽  
Outerplanar Graphs

List strong edge coloring of planar graphs with maximum degree 4

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

WEIGHTED-EDGE-COLORING OF k-DEGENERATE GRAPHS AND BIN-PACKING

2011 ◽  
Vol 12 (01n02) ◽  
pp. 109-124
Author(s):  
FLORIAN HUC
Keyword(s):  
Bin Packing ◽  
Edge Coloring ◽  
Weighted Graph ◽  
Packing Problem ◽  
Bipartite Graphs ◽  
Specific Class ◽  
Weighted Graphs ◽  
Outerplanar Graphs ◽  
Underlying Graph ◽  
Multiple Edges

The weighted-edge-coloring problem of an edge-weighted graph whose weights are between 0 and 1, consists in finding a coloring using as few colors as possible and satisfying the following constraints: the sum of weights of edges with the same color and incident to the same vertex must be at most 1. In 1991, Chung and Ross conjectured that if G is bipartite, then [Formula: see text] colors are always sufficient to weighted-edge-color (G,w), where [Formula: see text] is the maximum of the sums of the weights of the edges incident to a vertex. We prove this is true for edge-weighted graphs with multiple edges whose underlying graph is a tree. We further generalise this conjecture to non-bipartite graphs and prove the generalised conjecture for simple edge-weighted outerplanar graphs. Finally, we introduce a list version of this coloring together with the list-bin-packing problem, which allows us to obtain new results concerning the original coloring for a specific class of graphs, namely the k-weight-degenerate weighted graph.

Edge coloring by total labelings of outerplanar graphs

2013 ◽  
Vol 29 (11) ◽  
pp. 2129-2136
Author(s):  
Guang Hui Wang ◽  
Gui Ying Yan
Keyword(s):  
Edge Coloring ◽  
Outerplanar Graphs

Efficient Vertex- and Edge-Coloring of Outerplanar Graphs

1986 ◽  
Vol 7 (1) ◽  
pp. 131-136 ◽  
Author(s):  
Andrzej Proskurowski ◽  
Maciej M. Sysło
Keyword(s):  
Edge Coloring ◽  
Outerplanar Graphs

A polynomial-time nearly-optimal algorithm for an edge coloring problem in outerplanar graphs

2015 ◽  
Vol 65 (2) ◽  
pp. 351-367 ◽  
Author(s):  
Weifan Wang ◽  
Danjun Huang ◽  
Yanwen Wang ◽  
Yiqiao Wang ◽  
Ding-Zhu Du
Keyword(s):  
Polynomial Time ◽  
Optimal Algorithm ◽  
Edge Coloring ◽  
Outerplanar Graphs ◽  
Coloring Problem

scholarly journals Injective edge-coloring of graphs with given maximum degree

2021 ◽  
Vol 96 ◽  
pp. 103355
Author(s):  
Alexandr Kostochka ◽  
André Raspaud ◽  
Jingwei Xu
Keyword(s):  
Maximum Degree ◽  
Edge Coloring

scholarly journals Planar graphs with $\Delta \geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Marthe Bonamy ◽  
Benjamin Lévêque ◽  
Alexandre Pinlou
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Edge Coloring ◽  
Total Coloring ◽  
List Total Coloring ◽  
List Edge Coloring ◽  
International Audience

International audience For planar graphs, we consider the problems of <i>list edge coloring</i> and <i>list total coloring</i>. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total coloring is the problem of coloring the edges and the vertices while ensuring that two edges that are adjacent, two vertices that are adjacent, or a vertex and an edge that are incident receive different colors. In their list extensions, instead of having the same set of colors for the whole graph, every vertex or edge is assigned some set of colors and has to be colored from it. A graph is minimally edge or total choosable if it is list $\Delta$-edge-colorable or list $(\Delta +1)$-total-colorable, respectively, where $\Delta$ is the maximum degree in the graph. It is already known that planar graphs with $\Delta \geq 8$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable (Li Xu 2011), and that planar graphs with $\Delta \geq 7$ and no triangle sharing a vertex with a $C_4$ or no triangle adjacent to a $C_k (\forall 3 \leq k \leq 6)$ are minimally total colorable (Wang Wu 2011). We strengthen here these results and prove that planar graphs with $\Delta \geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable.

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