Graceful Labeling of Generalized Theta graphs

Sathiamoorthy, G.; Janakiraman, T. N.

doi:10.1007/s40009-018-0625-2

Cited by 5 publications

(3 citation statements)

References 2 publications

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“…Generalized Theta graphs, which have been widely studied for many graph theoretic problems (see e.g., [8,11,14,22,23,27]), are the main subject of two classical papers on the chromatic polynomial [10] and [28] which include the celebrated result that the zeros of the chromatic polynomials of the Generalized Theta graphs are dense in the whole complex plane with the possible exception of the unit disc around the origin (by including the join of Generalized Theta graphs with K 2 this extends to all of the complex plane). It is natural to study the DP color function of these graphs, independent of our motivating questions.…”

Section: Summary Of Resultsmentioning

confidence: 99%

On Polynomial Representations of the DP Color Function: Theta Graphs and Their Generalizations

Halberg¹,

Kaul²,

Liu³

et al. 2020

Preprint

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DP-coloring (also called correspondence coloring) is a generalization of list coloring that has been widely studied in recent years after its introduction by Dvořák and Postle in 2015. As the analogue of the chromatic polynomial P (G, m), the DP color function of a graph G, denoted P DP (G, m), counts the minimum number of DP-colorings over all possible m-fold covers. It is known that, unlike the list color function P ℓ (G, m), for any g ≥ 3 there exists a graph G with girth g such that P DP (G, m) < P (G, m) when m is sufficiently large. Thus, two fundamental open questions regarding the DP color function are: (i) for which G does there exist an N ∈ N such that P DP (G, m) = P (G, m) whenever m ≥ N , (ii) Given a graph G does there always exist an N ∈ N and a polynomial p(m) such that P DP (G, m) = p(m) whenever m ≥ N ?In this paper we give exact formulas for the DP color function of a Theta graph based on the parity of its path lengths. This gives an explicit answer, including the formulas for the polynomials that are not the chromatic polynomial, to both the questions above for Theta graphs. We extend this result to Generalized Theta graphs by characterizing the exact parity condition that ensures the DP color function eventually equals the chromatic polynomial. To answer the second question for Generalized Theta graphs, we confirm it for the larger class of graphs with a feedback vertex set of size one.

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Section: Summary Of Resultsmentioning

confidence: 99%

On Polynomial Representations of the DP Color Function: Theta Graphs and Their Generalizations

Halberg¹,

Kaul²,

Liu³

et al. 2020

Preprint

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“…We will assume that l 2 ≥ 2 when m ≥ 2 since we will only be considering simple graphs in this paper. 2 Studying list equitable total coloring of generalized theta graphs is quite natural as theta graphs and generalized theta graphs have many interesting properties that have been studied by many researchers (see [2,3,4,10,21,24,32]). For this paper, we prove a positive answer to the question: Does the LETCC hold for generalized theta graphs?…”

Section: Outline Of Results and An Open Questionmentioning

confidence: 99%

On List Equitable Total Colorings of the Generalized Theta Graph

Mudrock¹,

Marsh²,

Wagstrom³

2019

Preprint

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In 2003 Kostochka, Pelsmajer, and West introduced a list analogue of equitable coloring called equitable choosability. A k-assignment, L, for a graph G assigns a list, L(v), of k available colors to each v ∈ V (G), and an equitableand each color class of f has size at most ⌈|V (G)|/k⌉. In 2018, Kaul, Mudrock, and Pelsmajer subsequently introduced the List Equitable Total Coloring Conjecture which states that if T is a total graph of some simple graph, then T is equitably k-choosable for each k ≥ max{χ ℓ (T ), ∆(T )/2 + 2} where ∆(T ) is the maximum degree of a vertex in T and χ ℓ (T ) is the list chromatic number of T . In this paper we verify the List Equitable Total Coloring Conjecture for subdivisions of stars and the generalized theta graph.

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“…In this section, we compute the hyper Zagreb index of some generalized thetagraphs. A particular case of this structure can be found in [23].…”

Section: Hyper Zagreb Index Of Generalized Theta Graphsmentioning

confidence: 99%

On Hyper Zagreb Index of Certain Generalized Graph Structures

Wang¹,

Natarajan²,

Balachandran³

2020

Int. J. of Appl. Math.

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Let G = (V, E) be a graph with n vertices and m edges. The hyper Zagreb index of G, denoted by HM (G), is defined aswhere d G (v) denotes the degree of a vertex v in G. In this paper we compute the hyper Zagreb index of certain generalized graph structures such as generalized thorn graphs and generalized theta graphs. Also,for the first time, we determine exact values for hyper Zagreb index of some cycle related graphs, namely cycle with parallel P k chords, cycle with parallel C k chords and shell type graphs.

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Graceful Labeling of Generalized Theta graphs

Cited by 5 publications

References 2 publications

On Polynomial Representations of the DP Color Function: Theta Graphs and Their Generalizations

On Polynomial Representations of the DP Color Function: Theta Graphs and Their Generalizations

On List Equitable Total Colorings of the Generalized Theta Graph

On Hyper Zagreb Index of Certain Generalized Graph Structures

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