On the &lt;i&gt;r&lt;/i&gt;-dynamic coloring of subdivision-edge coronas of a path

Nandini, G.; Venkatachalam, M.; Ganfornina, Raúl Manuel Falcón

doi:10.3934/math.2020292

Cited by 7 publications

(3 citation statements)

References 17 publications

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“…Both concepts generalize in a natural way those classical ones concerning the proper coloring and the chromatic number of a graph, which arise indeed when r = 1. Since the original work of Montgomery, the case r > 1 has widely been dealt with in the literature for different types of graphs [8,6,7,11,12,14,15,19,22,24], with particular emphasis in the case r = 2 [1,2,3,4,5]. Of particular interest for the topic of this paper, let us remark the studies of Mohanapriya et al [20] and Furmańczyk et al [9] concerning the r-dynamic coloring of different families of helm graphs.…”

Section: Introductionmentioning

confidence: 99%

The $r$-dynamic edge coloring of a closed helm graph

Falcón,

Venkatachalam,

Gowri

et al. 2023

Revista De La Unión Matemática Argentina

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As a natural generalization of the classical coloring problem in graph theory, the dynamic coloring problem deals with the existence of a proper coloring c of a graph so that |c(N (v))| ≥ min{r, d(v)} for every vertex v. In this paper, we obtain the r-dynamic edge chromatic number of any given closed helm graph for any positive integer r. This coincides with the r-dynamic chromatic number of the line graph of a closed helm graph. 2020 Mathematics Subject Classification. 05C15.

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Section: Introductionmentioning

confidence: 99%

The $r$-dynamic edge coloring of a closed helm graph

Falcón,

Venkatachalam,

Gowri

et al. 2023

Revista De La Unión Matemática Argentina

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“…Computing the t-dynamic chromatic number of a graph constitutes the t-dynamic coloring problem for G. If t = 1, then it coincides with the classical coloring problem. Despite that the case t ≥ 2 has widely been dealt with in the literature, there exists only some partial results concerning corona products [40] and generalized corona products [41,42]. Particularly, Aparna and Mohanapriya [43] have recently dealt with the dynamic problem for extended neighborhood coronas whose center is a complete graph.…”

Section: Introductionmentioning

confidence: 99%

Optimal Shadow Allocations of Secret Sharing Schemes Arisen from the Dynamic Coloring of Extended Neighborhood Coronas

Ganfornina

Mohanapriya²,

Aparna³

2022

Mathematics

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Every t-dynamic proper n-coloring of a graph G describes a shadow allocation of any (n,t+1)-threshold secret sharing scheme based on G, so that, after just one round of communication, each participant can either reconstruct the secret, or obtain a different shadow from each one of his/her neighbors. Thus, for just one round of communication, this scheme is fair if and only if the threshold is either less than or equal to the minimum degree of G, or greater than or equal to its maximum degree. Despite that the dynamic coloring problem has widely been dealt with in the literature, a comprehensive study concerning this implementation in cryptography is still required. This paper delves into this topic by focusing on the use of extended neighborhood coronas for modeling communication networks whose average path lengths are small even after an asymptotic growth of their center and/or outer graphs. Particularly, the dynamic coloring problem is solved for any extended neighborhood corona with center path or star, for which we establish optimal shadow allocations of any (fair) threshold secret sharing scheme based on them. Some bounds are also established for the dynamic chromatic number of any extended neighborhood corona.

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“…Of particular relevance for the aim of this paper is the study of the rdynamic coloring of fan graphs and related families [23,26], a type of graphs for which the study of different proper colorings awakes interest in the literature [29,32,33]. This paper delves into this topic by focusing on the study of the r-dynamic chromatic number of four families of graphs: middle graphs, total graphs, central graphs and line graphs, derived all of them from a given fan graph F m,n .…”

Section: Introductionmentioning

confidence: 99%

On the r-dynamic coloring of some fan graph families

Ganfornina

Venkatachalam²,

Gowri³

et al. 2021

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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In this paper, we determine the r-dynamic chromatic number of the fan graph Fm,n and determine sharp bounds of this graph invariant for four related families of graphs: The middle graph M(Fm,n ), the total graph T (Fm,n ), the central graph C(Fm,n ) and the line graph L(Fm,n ). In addition, we determine the r-dynamic chromatic number of each one of these four families of graphs in case of being m = 1.

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On the <i>r</i>-dynamic coloring of subdivision-edge coronas of a path

Cited by 7 publications

References 17 publications

The $r$-dynamic edge coloring of a closed helm graph

The $r$-dynamic edge coloring of a closed helm graph

Optimal Shadow Allocations of Secret Sharing Schemes Arisen from the Dynamic Coloring of Extended Neighborhood Coronas

On the r-dynamic coloring of some fan graph families

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