List Edge-Coloring and Total Coloring in Graphs of Low Treewidth

2015 ◽  
Vol 81 (3) ◽  
pp. 272-282 ◽  
Author(s):  
Henning Bruhn ◽  
Richard Lang ◽  
Maya Stein
Keyword(s):  
Edge Coloring ◽  
Total Coloring ◽  
List 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.

scholarly journals On strong list edge coloring of subcubic graphs

2014 ◽  
Vol 333 ◽  
pp. 6-13 ◽  
Author(s):  
Hong Zhu ◽  
Zhengke Miao
Keyword(s):  
Edge Coloring ◽  
List Edge Coloring ◽  
Subcubic Graphs

scholarly journals Proof of the List Edge Coloring Conjecture for Complete Graphs of Prime Degree

10.37236/4084 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Group Action ◽  
Edge Coloring ◽  
Latin Squares ◽  
Combinatorial Nullstellensatz ◽  
Complete Graphs ◽  
Chromatic Index ◽  
Prime Degree ◽  
List Edge Coloring ◽  
Class 1

We prove that the list-chromatic index and paintability index of $K_{p+1}$ is $p$, for all odd primes $p$. This implies that the List Edge Coloring Conjecture holds for complete graphs with less then 10 vertices. It also shows that there are arbitrarily big complete graphs for which the conjecture holds, even among the complete graphs of class 1. Our proof combines the Quantitative Combinatorial Nullstellensatz with the Paintability Nullstellensatz and a group action on symmetric Latin squares. It displays various ways of using different Nullstellensätze. We also obtain a partial proof of a version of Alon and Tarsi's Conjecture about even and odd Latin squares.

scholarly journals Snarks with total chromatic number 5

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Gunnar Brinkmann ◽  
Myriam Preissmann ◽  
Diana Sasaki
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Total Coloring ◽  
Computer Search ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Total Chromatic Number ◽  
Vertex Number ◽  
Chordless Cycle

Graph Theory International audience A k-total-coloring of G is an assignment of k colors to the edges and vertices of G, so that adjacent and incident elements have different colors. The total chromatic number of G, denoted by χT(G), is the least k for which G has a k-total-coloring. It was proved by Rosenfeld that the total chromatic number of a cubic graph is either 4 or 5. Cubic graphs with χT = 4 are said to be Type 1, and cubic graphs with χT = 5 are said to be Type 2. Snarks are cyclically 4-edge-connected cubic graphs that do not allow a 3-edge-coloring. In 2003, Cavicchioli et al. asked for a Type 2 snark with girth at least 5. As neither Type 2 cubic graphs with girth at least 5 nor Type 2 snarks are known, this is taking two steps at once, and the two requirements of being a snark and having girth at least 5 should better be treated independently. In this paper we will show that the property of being a snark can be combined with being Type 2. We will give a construction that gives Type 2 snarks for each even vertex number n≥40. We will also give the result of a computer search showing that among all Type 2 cubic graphs on up to 32 vertices, all but three contain an induced chordless cycle of length 4. These three exceptions contain triangles. The question of the existence of a Type 2 cubic graph with girth at least 5 remains open.

scholarly journals Super local edge anti-magic total coloring of paths and its derivation

2020 ◽  
Vol 3 (2) ◽  
pp. 126
Author(s):  
Fawwaz Fakhrurrozi Hadiputra ◽  
Denny Riama Silaban ◽  
Tita Khalis Maryati
Keyword(s):  
Chromatic Number ◽  
Minimum Weight ◽  
Edge Coloring ◽  
Simple Graph ◽  
Total Coloring ◽  
Local Edge

<p>Suppose <em>G</em>(<em>V,E</em>) be a connected simple graph and suppose <em>u,v,x</em> be vertices of graph <em>G</em>. A bijection <em>f</em> : <em>V</em> ∪ <em>E</em> → {1,2,3,...,|<em>V</em> (<em>G</em>)| + |<em>E</em>(<em>G</em>)|} is called super local edge antimagic total labeling if for any adjacent edges <em>uv</em> and <em>vx</em>, <em>w</em>(<em>uv</em>) 6= <em>w</em>(<em>vx</em>), which <em>w</em>(<em>uv</em>) = <em>f</em>(<em>u</em>)+<em>f</em>(<em>uv</em>)+<em>f</em>(<em>v</em>) for every vertex <em>u,v,x</em> in <em>G</em>, and <em>f</em>(<em>u</em>) &lt; <em>f</em>(<em>e</em>) for every vertex <em>u</em> and edge <em>e</em> ∈ <em>E</em>(<em>G</em>). Let γ(<em>G</em>) is the chromatic number of edge coloring of a graph <em>G</em>. By giving <em>G</em> a labeling of <em>f</em>, we denotes the minimum weight of edges needed in <em>G</em> as γ<em>leat</em>(<em>G</em>). If every labels for vertices is smaller than its edges, then it is be considered γ<em>sleat</em>(<em>G</em>). In this study, we proved the γ sleat of paths and its derivation.</p>

Optimal channel assignment with list-edge coloring

2019 ◽  
Vol 38 (1) ◽  
pp. 197-207 ◽  
Author(s):  
Huijuan Wang ◽  
Panos M. Pardalos ◽  
Bin Liu
Keyword(s):  
Channel Assignment ◽  
Edge Coloring ◽  
List Edge Coloring

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
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