Total colorings-a survey

Geetha, J.; Narayanan, Narayanan; Somasundaram, K.

doi:10.1080/09728600.2023.2187960

Cited by 4 publications

(2 citation statements)

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“…The Total Coloring Conjecture (TCC) (Behzad [3] and Vizing [12] independently proposed) is a famous open problem in graph theory that states that any simple graph G with maximum degree ∆(G) can be totally colored with at most ∆(G) + 2 colors. The progress on this conjecture had been slow in the initial years but several advancements are in progress [8]. The graphs that can be totally colored in ∆(G) + 1 colors are said to be type I graphs and the graph with the total chromatic number ∆(G) + 2 is said to be type II.…”

Section: Introductionmentioning

confidence: 99%

Total chromatic number for some classes of Cayley graphs

2023

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Section: Introductionmentioning

confidence: 99%

Total chromatic number for some classes of Cayley graphs

2023

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“…The total coloring conjecture has also been verified for several other classes of graphs. Some good survey articles and books listing the basic techniques and other results on total coloring are Yap [5], Borodin, [6], and Geetha et al [7].…”

Section: Introductionmentioning

confidence: 99%

Total Coloring of Some Classes of Cayley Graphs on Non-Abelian Groups

et al. 2022

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Total Coloring of a graph G is a type of graph coloring in which any two adjacent vertices, an edge, and its incident vertices or any two adjacent edges do not receive the same color. The minimum number of colors required for the total coloring of a graph is called the total chromatic number of the graph, denoted by χ″(G). Mehdi Behzad and Vadim Vizing simultaneously worked on the total colorings and proposed the Total Coloring Conjecture (TCC). The conjecture states that the maximum number of colors required in a total coloring is Δ(G)+2, where Δ(G) is the maximum degree of the graph G. Graphs derived from the symmetric groups are robust graph structures used in interconnection networks and distributed computing. The TCC is still open for the circulant graphs. In this paper, we derive the upper bounds for χ″(G) of some classes of Cayley graphs on non-abelian groups, typically Cayley graphs on the symmetric groups and dihedral groups. We also obtain the upper bounds of the total chromatic number of complements of Kneser graphs.

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Total Coloring of Some Graph Operations

Kavaskar,

Sukumaran

2024

Lecture Notes in Computer Science

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Total colorings-a survey

Cited by 4 publications

References 130 publications

Total chromatic number for some classes of Cayley graphs

Total chromatic number for some classes of Cayley graphs

Total Coloring of Some Classes of Cayley Graphs on Non-Abelian Groups

Total Coloring of Some Graph Operations

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