An introduction to the <i>k</i>-defect polynomials

Mphako-Banda, Eunice

doi:10.2989/16073606.2018.1443168

Cited by 10 publications

(10 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [3], the k-defect polynomial of a graph G is defined as the polynomial which determines the number of λ-colourings which result in k bad edges. Clearly, the notion of δ (k) -colouring is a derivative of k-defect colouring.…”

Section: Discussionmentioning

confidence: 99%

“…The motivation for the number range of colours is that δ (k) -colouring has a relation with k-defect colouring (see [3]). A trivial observation is that for a colouring C = {c 1 } or for a colouring C = {c i : 1 ≤ i ≤ k, k ≥ 2} and any colour c j ∈ C , a defect colouring of graph G such that all edges are bad is easily possible through a δ (1) -colouring.…”

Section: δ (K) -Colouring Of Graphsmentioning

confidence: 99%

“…(iii) The rim vertices v , 1 ≤ i ≤ n − 1 can be properly coloured with two colours in 3 2 ways. The vertices v 0 , v n can be coloured with the third colour to result in 1 bad edge as a minimum.…”

Section: δ (K) -Colouring Of Graphsmentioning

confidence: 99%

“…In [3], it is claimed that at the time of writing only two explicit expressions of k-defect polynomials are known (for trees and cycle graphs). We add an initial result for determining b k (K n ).…”

mentioning

confidence: 99%

See 3 more Smart Citations

$δ^{(k)}$-Colouring of Cycle Related Graphs

Kok,

Naduvath

2018

Preprint

View full text Add to dashboard Cite

With respect to a proper colouring of a graph G, we know that δ(G) ≤ χ(G) ≤ ∆(G) + 1. If distinct colours represent distinct technology types to be located at vertices the question arises on how to place at least one of each of k, 1 ≤ k < χ(G) technology types together with the minimum adjacency between similar technology types. In an improper colouring an edge uv such that c(u) = c(v) is called a bad edge. In this paper, we introduce the notion of δ (k) -colouring which is a near proper colouring of G with exactly 1 ≤ k < χ(G) distinct colours which minimizes the number of bad edges.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: δ (K) -Colouring Of Graphsmentioning

confidence: 99%

Section: δ (K) -Colouring Of Graphsmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

$δ^{(k)}$-Colouring of Cycle Related Graphs

Kok,

Naduvath

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…2 In an improper coloring an edge uv for which, c(u) = c(v) is called a bad edge. See [5] for an introduction to defect colorings of graphs. It is observed that the number of edges of G which are omitted from E χ is the minimum number of bad edges in a bad chromatic completion of a graph G.…”

Section: Introductionmentioning

confidence: 99%

Heuristic method to determine lucky k-polynomials for k-colorable graphs

Kok

2019

Acta Universitatis Sapientiae, Informatica

View full text Add to dashboard Cite

The existence of edges is a huge challenge with regards to determining lucky k-polynomials of simple connected graphs in general. In this paper the lucky 3-polynomials of path and cycle graphs of order, 3 ≤ n ≤ 8 are presented as the basis for the heuristic method to determine the lucky k-polynomials for k-colorable graphs. The difficulty of adjacency with graphs is illustrated through these elementary graph structures. The results are also illustratively compared with the results for null graphs (edgeless graphs). The paper could serve as a basis for finding recurrence results through innovative methodology.

show abstract

On δ(k)-Coloring of Some Cycle-Related Graphs

Ellumkalayil,

Naduvath

2023

J. Inter. Net.

View full text Add to dashboard Cite

A graph coloring is proper when the colors assigned to a pair of adjacent vertices in it are different and it is improper when at least one of the adjacent pair of vertices receives the same color. When the minimum number of colors required in a proper coloring of a graph is not available, coloring the graph with the available colors, say [Formula: see text] colors, will lead at least an edge to have its end vertices colored with a same color. Such an edge is called a bad edge. In a proper coloring of a graph [Formula: see text], every color class is an independent set. However, in an improper coloring there can be few color classes that are non-independent. In this paper, we use the concept of [Formula: see text]-coloring, which permits only one color class to be non-independent and determine the minimum number of bad edges, which is denoted by [Formula: see text], obtained from the same for some cycle-related graphs.

show abstract

An introduction to the k-defect polynomials

Cited by 10 publications

References 4 publications

$δ^{(k)}$-Colouring of Cycle Related Graphs

$δ^{(k)}$-Colouring of Cycle Related Graphs

Heuristic method to determine lucky k-polynomials for k-colorable graphs

On δ(k)-Coloring of Some Cycle-Related Graphs

Contact Info

Product

Resources

About