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

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

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

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

scholarly journals Acyclic list edge coloring of outerplanar graphs

2013 ◽  
Vol 313 (3) ◽  
pp. 301-311
Author(s):  
Qiaojun Shu ◽  
Yiqiao Wang ◽  
Weifan Wang
Keyword(s):  
Edge Coloring ◽  
Outerplanar Graphs ◽  
List Edge Coloring

Injective edge-coloring of subcubic graphs

2021 ◽  
Author(s):  
Baya Ferdjallah ◽  
Samia Kerdjoudj ◽  
André Raspaud
Keyword(s):  
Edge Coloring ◽  
Upper Bounds ◽  
Chromatic Index ◽  
Radio Network ◽  
Outerplanar Graphs ◽  
Maximum Average Degree ◽  
Injective Coloring ◽  
Packet Radio Network ◽  
Minimum Integer ◽  
Subcubic Graphs

An injective edge-coloring [Formula: see text] of a graph [Formula: see text] is an edge-coloring such that if [Formula: see text], [Formula: see text], and [Formula: see text] are three consecutive edges in [Formula: see text] (they are consecutive if they form a path or a cycle of length three), then [Formula: see text] and [Formula: see text] receive different colors. The minimum integer [Formula: see text] such that, [Formula: see text] has an injective edge-coloring with [Formula: see text] colors, is called the injective chromatic index of [Formula: see text] ([Formula: see text]). This parameter was introduced by Cardoso et al. [Injective coloring of graphs, Filomat 33(19) (2019) 6411–6423, arXiv:1510.02626] motivated by the Packet Radio Network problem. They proved that computing [Formula: see text] of a graph [Formula: see text] is NP-hard. We give new upper bounds for this parameter and we present the relationships of the injective edge-coloring with other colorings of graphs. We study the injective edge-coloring of some classes of subcubic graphs. We prove that a subcubic bipartite graph has an injective chromatic index bounded by [Formula: see text]. We also prove that if [Formula: see text] is a subcubic graph with maximum average degree less than [Formula: see text] (respectively, [Formula: see text]), then [Formula: see text] admits an injective edge-coloring with at most 4 (respectively, [Formula: see text]) colors. Moreover, we establish a tight upper bound for subcubic outerplanar graphs.

A STUDY ON FUZZY EQUITABLE EDGE COLORING OF WHEEL RELATED GRAPHS

2020 ◽  
Vol 9 (11) ◽  
pp. 9311-9317
Author(s):  
K. Sivaraman ◽  
R.V. Prasad
Keyword(s):  
Edge Coloring ◽  
Graph Labeling ◽  
Equitable Edge Coloring

Equitable edge coloring is a kind of graph labeling with the following restrictions. No two adjacent edges receive same label (color). and number of edges in any two color classes differ by at most one. In this work we are going to present the Fuzzy equitable edge coloring of some wheel related graphs.

scholarly journals Metric Dimension Parameterized By Treewidth

Algorithmica ◽  
2021 ◽  
Author(s):  
Édouard Bonnet ◽  
Nidhi Purohit
Keyword(s):  
Computable Function ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Input Graph ◽  
Outerplanar Graphs ◽  
Bounded Treewidth ◽  
Fpt Algorithm

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

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