scholarly journals Odd-Graceful Total Colorings for Constructing Graphic Lattice

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 109
Author(s):  
Jing Su ◽  
Hui Sun ◽  
Bing Yao
Keyword(s):  
Graph Isomorphism ◽  
Total Coloring ◽  
Connected Graphs ◽  
Linearly Independent ◽  
Total Coloring Conjecture

The security of passwords generated by the graphic lattices is based on the difficulty of the graph isomorphism, graceful tree conjecture, and total coloring conjecture. A graphic lattice is generated by a graphic base and graphical operations, where a graphic base is a group of disjointed, connected graphs holding linearly independent properties. We study the existence of graphic bases with odd-graceful total colorings and show graphic lattices by vertex-overlapping and edge-joining operations; we prove that these graphic lattices are closed to the odd-graceful total coloring.

scholarly journals Total Coloring Conjecture for Certain Classes of Graphs

Algorithms ◽  
2018 ◽  
Vol 11 (10) ◽  
pp. 161 ◽  
Author(s):  
R. Vignesh ◽  
J. Geetha ◽  
K. Somasundaram
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Line Graph ◽  
Product Line ◽  
Total Coloring ◽  
Lexicographic Product ◽  
Total Chromatic Number ◽  
Minimum Number ◽  
Total Coloring Conjecture

A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no two adjacent or incident elements receive the same color. The total chromatic number of a graph G, denoted by χ ′ ′ ( G ) , is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G, Δ ( G ) + 1 ≤ χ ′ ′ ( G ) ≤ Δ ( G ) + 2 , where Δ ( G ) is the maximum degree of G. In this paper, we prove the total coloring conjecture for certain classes of graphs of deleted lexicographic product, line graph and double graph.

Total colorings of circulant graphs

2020 ◽  
pp. 2150050
Author(s):  
J. Geetha ◽  
K. Somasundaram ◽  
Hung-Lin Fu
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Total Coloring ◽  
Circulant Graphs ◽  
Total Chromatic Number ◽  
Number Formula ◽  
Total Coloring Conjecture

The total chromatic number [Formula: see text] is the least number of colors needed to color the vertices and edges of a graph [Formula: see text] such that no incident or adjacent elements (vertices or edges) receive the same color. Behzad and Vizing proposed a well-known total coloring conjecture (TCC): [Formula: see text], where [Formula: see text] is the maximum degree of [Formula: see text]. For the powers of cycles, Campos and de Mello proposed the following conjecture: Let [Formula: see text] denote the graphs of powers of cycles of order [Formula: see text] and length [Formula: see text] with [Formula: see text]. Then, [Formula: see text] In this paper, we prove the Campos and de Mello’s conjecture for some classes of powers of cycles. Also, we prove the TCC for complement of powers of cycles.

The Smarandachely Adjacent Vertex Distinguishing E-Total Coloring of some Join Graphs

2013 ◽  
Vol 475-476 ◽  
pp. 379-382
Author(s):  
Mu Chun Li ◽  
Shuang Li Wang ◽  
Li Li Wang
Keyword(s):  
Chromatic Number ◽  
Total Coloring ◽  
Adjacent Vertex ◽  
Analysis Method ◽  
Total Chromatic Number ◽  
Join Graph ◽  
Total Coloring Conjecture

Using the analysis method and the function of constructing the Smarandachely adjacent vertex distinguishing E-total coloring function, the Smarandachely adjacent vertex distinguishing E-total coloring of join graphs are mainly discussed, and the Smarandachely adjacent vertex distinguishing E-total chromatic number of join graph are obtained. The Smarandachely adjacent vertex distinguishing E-total coloring conjecture is further validated.

Tight Upper Bound of the 3-Total-Rainbow Index for 2-(Edge-)Connected Graphs

2021 ◽  
pp. 2142001
Author(s):  
Yingbin Ma ◽  
Wenhan Zhu
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Total Coloring ◽  
Colored Graph ◽  
Fixed Integer ◽  
Connected Graphs ◽  
Vertex Set ◽  
Rainbow Coloring

Let [Formula: see text] be an edge-colored graph with order [Formula: see text] and [Formula: see text] be a fixed integer satisfying [Formula: see text]. For a vertex set [Formula: see text] of at least two vertices, a tree containing the vertices of [Formula: see text] in [Formula: see text] is called an [Formula: see text]-tree. The [Formula: see text]-tree [Formula: see text] is a total-rainbow [Formula: see text]-tree if the elements of [Formula: see text], except for the vertex set [Formula: see text], have distinct colors. A total-colored graph [Formula: see text] is said to be total-rainbow [Formula: see text]-tree connected if for every set [Formula: see text] of [Formula: see text] vertices in [Formula: see text], there exists a total-rainbow [Formula: see text]-tree in [Formula: see text], while the total-coloring of [Formula: see text] is called a [Formula: see text]-total-rainbow coloring. The [Formula: see text]-total-rainbow index of a nontrivial connected graph [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed in a [Formula: see text]-total-rainbow coloring of [Formula: see text]. In this paper, we show a sharp upper bound for [Formula: see text], where [Formula: see text] is a 2-connected or 2-edge-connected graph.

scholarly journals A Sufficient Condition for Planar Graphs of Maximum Degree 6 to be Totally 7-Colorable

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Enqiang Zhu ◽  
Yongsheng Rao
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Simple Graph ◽  
Total Coloring ◽  
Sufficient Condition ◽  
Open Case ◽  
Total Coloring Conjecture

A total k-coloring of a graph is an assignment of k colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The total coloring conjecture (TCC) states that every simple graph G has a total ΔG+2-coloring, where ΔG is the maximum degree of G. This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph G of ΔG≥9 or ΔG∈7,8 with some restrictions has a total ΔG+1-coloring. In particular, in (Shen and Wang, 2009), the authors proved that every planar graph with maximum degree 6 and without 4-cycles has a total 7-coloring. In this paper, we improve this result by showing that every diamond-free and house-free planar graph of maximum degree 6 is totally 7-colorable if every 6-vertex is not incident with two adjacent four cycles or three cycles of size p,q,ℓ for some p,q,ℓ∈3,4,4,3,3,4.

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.

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.

Zero-divisor graphs and total coloring conjecture

2020 ◽  
Vol 24 (24) ◽  
pp. 18273-18285
Author(s):  
Nilesh Khandekar ◽  
Vinayak Joshi
Keyword(s):  
Zero Divisor ◽  
Total Coloring ◽  
Total Coloring Conjecture

Total coloring of the prismatic graphs

2020 ◽  
Vol 12 (03) ◽  
pp. 2050032
Author(s):  
S. Mohan ◽  
K. Somasundaram
Keyword(s):  
Total Coloring ◽  
Tight Bound ◽  
Total Coloring Conjecture

A total coloring of a graph is an assignment of colors to all the elements of the graph such that no two adjacent or incident elements receive the same color. A graph [Formula: see text] is prismatic, if for every triangle [Formula: see text], every vertex not in [Formula: see text] has exactly one neighbor in [Formula: see text]. In this paper, we prove the total coloring conjecture (TCC) for prismatic graphs and the tight bound of the TCC for some classes of prismatic graphs.

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