Total coloring of quasi-line graphs and inflated graphs

Mohan, S.; Geetha, J.; Somasundaram, K.

doi:10.1142/s1793830921500609

Cited by 3 publications

(1 citation statement)

References 15 publications

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“…In 2018, Vignesh et al [12] conjectured that all line graphs of complete graphs 𝐿(𝐾 𝑛 ) are Type 1. In 2021, Mohan et al [8] verified the TCC to the set of quasi-line graphs, which is a generalization of line graphs, and presented some infinite families of Type 1 graphs. In 2022, Jayaraman et al [6] determined the total chromatic number for certain line graphs.…”

Section: Introductionmentioning

confidence: 99%

On the conformability of regular line graphs

Faria,

Nigro,

Sasaki

2023

RAIRO-Oper. Res.

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Let $G=(V,E)$ be a graph and the \emph{deficiency of $G$} be $def(G)=\sum_{v \in V(G)} (\Delta(G)-d_{G}(v))$, where $d_{G}(v)$ is the degree of a vertex $v$ in $G$. A vertex coloring $\varphi :V(G)\to \{1,2,...,\Delta(G)+1\}$ is called \emph{conformable} if the number of color classes (including empty color classes) of parity different from that of $|V(G)|$ is at most $def(G)$. A general characterization for conformable graphs is unknown. Conformability plays a key role in the total chromatic number theory. It is known that if $G$ is \textit{Type~1}, then $G$ is conformable. In this paper, we prove that if $G$ is $k$-regular and \textit{Class~1}, then $L(G)$ is conformable. As an application of this statement we establish that the line graph of complete graph $L(K_n)$ is conformable, which is a positive evidence towards the Vignesh et al.'s conjecture that $L(K_n)$ is \textit{Type~1}.

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Section: Introductionmentioning

confidence: 99%

On the conformability of regular line graphs

Faria,

Nigro,

Sasaki

2023

RAIRO-Oper. Res.

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Total colorings-a survey

Geetha

Narayanan

Somasundaram

2023

AKCE International Journal of Graphs and Combinatorics

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The line graphs of Möbius ladder graphs are Type 1

Faria,

Nigro,

Sasaki

2024

Anais Do IX Encontro De Teoria Da Computação (ETC 2024)

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A k-total coloring of G is an assignment of k colors to its elements (vertices and edges) such that adjacent or incident elements have distinct colors. The total chromatic number of a graph G is the smallest integer k for which G has a k-total coloring. If the total chromatic number of G is ∆(G) + 1, then we say that G is Type 1. The line graph of G, denoted by L(G), is the graph whose vertex set is the edge set of G and two vertices of the line graph of G are adjacent if the corresponding edges are adjacent in G. In this paper, we prove that the line graphs of Möbius ladder graphs, L(M2n), are Type 1.

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Total coloring of quasi-line graphs and inflated graphs

Cited by 3 publications

References 15 publications

On the conformability of regular line graphs

On the conformability of regular line graphs

Total colorings-a survey

The line graphs of Möbius ladder graphs are Type 1

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