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>

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.

2-distance vertex-distinguishing total coloring of graphs

2018 ◽  
Vol 10 (02) ◽  
pp. 1850018
Author(s):  
Yafang Hu ◽  
Weifan Wang
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Simple Graph ◽  
Total Coloring ◽  
Total Chromatic Number ◽  
Minimum Number ◽  
Number Formula ◽  
Proper Total Coloring ◽  
Distance Formula

A [Formula: see text]-distance vertex-distinguishing total coloring of a graph [Formula: see text] is a proper total coloring of [Formula: see text] such that any pair of vertices at distance [Formula: see text] have distinct sets of colors. The [Formula: see text]-distance vertex-distinguishing total chromatic number [Formula: see text] of [Formula: see text] is the minimum number of colors needed for a [Formula: see text]-distance vertex-distinguishing total coloring of [Formula: see text]. In this paper, we determine the [Formula: see text]-distance vertex-distinguishing total chromatic number of some graphs such as paths, cycles, wheels, trees, unicycle graphs, [Formula: see text], and [Formula: see text]. We conjecture that every simple graph [Formula: see text] with maximum degree [Formula: see text] satisfies [Formula: see text].

Total coloring conjecture for vertex, edge and neighborhood corona products of graphs

2019 ◽  
Vol 11 (01) ◽  
pp. 1950014
Author(s):  
Radhakrishnan Vignesh ◽  
Jayabalan Geetha ◽  
Kanagasabapathi Somasundaram
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Simple Graph ◽  
Total Coloring ◽  
Tight Bound ◽  
Total Chromatic Number ◽  
Minimum Number ◽  
Total Coloring Conjecture

A total coloring of a graph [Formula: see text] is an assignment of colors to the elements of the graph [Formula: see text] such that no adjacent vertices and edges receive the same color. The total chromatic number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any simple graph [Formula: see text], [Formula: see text], where [Formula: see text] is the maximum degree of [Formula: see text]. In this paper, we prove the tight bound of the total coloring conjecture for the three types of corona products (vertex, edge and neighborhood) of graphs.

scholarly journals The Dominator Edge Coloring of Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Minhui Li ◽  
Shumin Zhang ◽  
Caiyun Wang ◽  
Chengfu Ye
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Simple Graph ◽  
Edge Colorings ◽  
Color Class ◽  
Proper Edge Coloring ◽  
Minimum Number ◽  
Relationship Of ◽  
The Relationship

Let G be a simple graph. A dominator edge coloring (DE-coloring) of G is a proper edge coloring in which each edge of G is adjacent to every edge of some color class (possibly its own class). The dominator edge chromatic number (DEC-number) of G is the minimum number of color classes among all dominator edge colorings of G , denoted by χ d ′ G . In this paper, we establish the bounds of the DEC-number of a graph, present the DEC-number of special graphs, and study the relationship of the DEC-number between G and the operations of G .

A Note on the Adjacent Vertex Distinguishing Total Chromatic Number of Graph

2011 ◽  
Vol 474-476 ◽  
pp. 2341-2345
Author(s):  
Zhi Wen Wang
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Simple Graph ◽  
Total Coloring ◽  
Adjacent Vertex ◽  
Total Chromatic Number ◽  
Parallel Graph ◽  
Total Coloring Conjecture

A total coloring of a simple graph G is called adjacent vertex distinguishing if for any two adjacent and distinct vertices u and v in G, the set of colors assigned to the vertices and the edges incident to u differs from the set of colors assigned to the vertices and the edges incident to v. In this paper we shall prove the series-parallel graph with maximum degree 3 and the series-parallel graph whose the number of edges is the double of maximum degree minus 1 satisfy the adjacent vertex distinguishing total coloring conjecture.

Vertex-Distinguishing Total Coloring of Ladder Graphs (N=0(mod 8) )

2011 ◽  
Vol 225-226 ◽  
pp. 243-246
Author(s):  
Zhi Wen Wang
Keyword(s):  
Chromatic Number ◽  
Simple Graph ◽  
Total Coloring ◽  
Total Chromatic Number ◽  
Proper Total Coloring ◽  
Ladder Graphs

A proper total coloring of a simple graph G is called vertex distinguishing if for any two distinct vertices u and v in G, the set of colors assigned to the elements incident to u differs from the set of colors incident to v. The minimal number of colors required for a vertex distinguishing total coloring of G is called the vertex distinguishing total coloring chromatic number. In a paper, we give a “triangle compositor”, by the compositor, we proved that when n=0(mod 8) and , vertex distinguishing total chromatic number of “ladder graphs” is n.

Adjacent Vertex-Distinguishing E-Total Coloring on the Multiple Join Graph of Several Kinds of Particular Graphs

2012 ◽  
Vol 546-547 ◽  
pp. 489-494
Author(s):  
Mu Chun Li ◽  
Li Zhang
Keyword(s):  
Positive Integer ◽  
Chromatic Number ◽  
Simple Graph ◽  
Total Coloring ◽  
Adjacent Vertex ◽  
Total Chromatic Number ◽  
Join Graph ◽  
Uv C

Let G(V,E) be a simple graph, k be a positive integer, f be a mapping from V(G)E(G) to 1,2,...k. If uvE(G), we have f(u)≠f(v),f(u)≠f(uv) ,f(v)≠f(uv) ,C(u)≠C(v) , where C(u). Then f is called the adjacent vertex-distinguishing E-total coloring of G. The number is called the adjacent vertex –distinguishing E-total chromatic number of G. In this paper, the adjacent vertex –distinguishing E-total chromatic number of the multiple join graph of several kinds of particular graphs is discussed by using construct coloring function.

Adjacent Vertex-Distinguishing E-Total Coloring on the Multiple Join Graph of Complete Graph and Wheel

2013 ◽  
Vol 303-306 ◽  
pp. 1605-1608
Author(s):  
Mu Chun Li ◽  
Li Zhang
Keyword(s):  
Positive Integer ◽  
Chromatic Number ◽  
Complete Graph ◽  
Simple Graph ◽  
Total Coloring ◽  
Adjacent Vertex ◽  
Chromatic Numbers ◽  
Total Chromatic Number ◽  
Join Graph ◽  
Uv C

υυυLet G be a simple graph, k be a positive integer, f be a mapping from V(G)∪E(G) to {1,2,...,k} . If ∀uv∈E(G) , we have f(u)≠f(v) , f(u)≠f(uv),f(v)≠f(uv) , C(u)≠C(v), where C(u)={f(u)}∪{f(uv)|uv∈E(G)}. Then f is called the adjacent vertex distinguishing E-total coloring of G. The number is called the adjacent vertex –distinguishing E-total chromatic number of χSubscript text(G)=min{k|G has a k-AVDETC} . The adjacent vertex distinguishing E-total chromatic numbers of the multiple join graph of wheel and complete graph are obtained in this paper.

scholarly journals On The Local Edge Antimagic Coloring of Corona Product of Path and Cycle

CAUCHY ◽  
2019 ◽  
Vol 6 (1) ◽  
pp. 40
Author(s):  
Siti Aisyah ◽  
Ridho Alfarisi ◽  
Rafiantika M. Prihandini ◽  
Arika Indah Kristiana ◽  
Ratna Dwi Christyanti
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Edge Coloring ◽  
Antimagic Labeling ◽  
Proper Edge Coloring ◽  
Minimum Number ◽  
Vertex Set ◽  
Local Edge ◽  
Corona Product

<p>Let  be a nontrivial and connected graph of vertex set  and edge set  . A bijection  is called a local edge antimagic labeling if for any two adjacent edges  and , where for . Thus, the local edge antimagic labeling induces a proper edge coloring of G if each edge e assigned the color  . The color of each an edge <em>e</em> = <em>uv</em> is assigned bywhich is defined by the sum of label both and vertices  and  . The local edge antimagic chromatic number, denoted by  is the minimum number of colors taken over all colorings induced by local edge antimagic labeling of   . In our paper, we present the local edge antimagic coloring of corona product of path and cycle, namely path corona cycle, cycle corona path, path corona path, cycle corona cycle.</p><p><strong>Keywords:</strong> Local antimagic; edge coloring; corona product; path; cycle.</p>

scholarly journals Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 293
Author(s):  
Xinyue Liu ◽  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao
Keyword(s):  
Linear Time ◽  
Minimum Weight ◽  
Domination Number ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Domination Problem ◽  
Np Complete ◽  
Chordal Bipartite Graphs ◽  
Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

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