Total Coloring Conjecture for Certain Classes of Graphs

Vignesh, Radhakrishnan; Geetha, J.; Somasundaram, K.

doi:10.3390/a11100161

Cited by 10 publications

(7 citation statements)

References 18 publications

(21 reference statements)

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“…In this paper, we establish a relationship between the chromatic index of 𝐺 and the conformability of 𝐿(𝐺), proving that if 𝐺 is 𝑘-regular Class 1, then 𝐿(𝐺) is conformable. We apply this statement to prove that 𝐿(𝐾 𝑛 ) is conformable, supporting the Vignesh et al's [12] conjecture. In addition, we propose a question about the existence of 𝐿(𝐺) non-conformable of a 𝑘-regular 𝐺 and investigate this question, by presenting non-conformable graphs which are not line graphs.…”

Section: Introductionsupporting

confidence: 77%

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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: Introductionsupporting

confidence: 77%

“…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.…”

Section: Introductionmentioning

confidence: 99%

On the conformability of regular line graphs

Faria,

Nigro,

Sasaki

2023

RAIRO-Oper. Res.

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“…The objective function (41) includes the relation specified in (40). Constraints (42) ensure that the subgraph induced by the variables x v and y e is a union of disjoint cycles, since every node has either degree zero or two.…”

Section: Discussionmentioning

confidence: 99%

“…Hence, if the conjecture were true, we would be left with the NP-Complete problem of deciding whether χ T (G) = ∆(G) + 1. While the conjecture is valid for specific classes of graphs (e.g., see [41]), the conjecture is still open for general graphs 1 . The Total Coloring Problem generalizes both the Vertex Coloring Problem, where we have to color only the vertices of G, and the Edge Coloring Problem, where instead we have to color only the edges.…”

Section: Introductionmentioning

confidence: 99%

Total Coloring and Total Matching: Polyhedra and Facets

Ferrarini¹,

Gualandi²

2021

Preprint

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A total coloring of a graph G = (V, E) is an assignment of colors to vertices and edges such that neither two adjacent vertices nor two incident edges get the same color, and, for each edge, the end-points and the edge itself receive different colors. Any valid total coloring induces a partition of the elements of G into total matchings, which are defined as subsets of vertices and edges that can take the same color. In this paper, we propose Integer Linear Programming models for both the Total Coloring and the Total Matching problems, and we study the strength of the corresponding Linear Programming relaxations. The total coloring is formulated as the problem of finding the minimum number of total matchings that cover all the graph elements, and we prove that this relaxation is tighter than a natural assignment model. This covering formulation can be solved by a column generation algorithm, where the pricing subproblem corresponds to the Weighted Total Matching Problem. Hence, we study the Total Matching Polytope. We introduce two families of nontrivial valid inequalities: congruent-2k3 cycle inequalities based on the parity of the vertex set induced by the cycle, and clique inequalities induced by complete subgraphs of even order. We prove that congruent-2k3 cycle inequalities are facet-defining only when k = 4, while the even cliques are always facet-defining. Since the separation problem of the clique inequalities of even order is NP-hard, we get a polyhedral proof of the NP-hardness of the Weighted Total Matching Problem.

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“…Direct product, cartesian product, strong product and lexicographic product graphs given by Imrich [8] et la. Recently, Vignesh et al [21,16] verified TCC for certain classes of deleted lexicogaphic product graphs. In [20], they also proved that Vertex, Edge and Neighborhood corona products of graphs are type-I graphs.…”

Section: Introductionmentioning

confidence: 99%

Total coloring conjecture on certain classes of product graphs

Somasundaram¹,

Geetha²,

Vignesh³

2023

EJGTA

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A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no adjacent vertices and edges 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, ∆(G) + 1 ≤ χ ′′ (G) ≤ ∆(G) + 2, where ∆(G) is the maximum degree of G. In this paper, we prove the Behzad and Vizing conjecture for Indu -Bala product graph, Skew and Converse Skew product graph, Cover product graph, Clique cover product graph and Comb product graph.

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Total Coloring Conjecture for Certain Classes of Graphs

Cited by 10 publications

References 18 publications

On the conformability of regular line graphs

On the conformability of regular line graphs

Total Coloring and Total Matching: Polyhedra and Facets

Total coloring conjecture on certain classes of product graphs

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