Malaya Journal of Matematik, Vol. 8, No. 3, 767-774, 2020
https://doi.org/10.26637/MJM0803/0006
Lucky χ -polynomial of graphs of order 5
Johan Kok
Abstract
The concept of Lucky k-polynomials was recently introduced for null and complete split graphs. This paper
extends on the introductory work and presents Lucky χ -polynomials (k = χ (G)) for graphs of order 5. The
methodical work done for graphs of order 5 serves mainly to set out the fundamental method to be used for all
other classes of graphs. Finally, further problems for research related to this concept are presented.
Keywords
Chromatic completion number, chromatic completion graph, chromatic completion edge, bad edge, Lucky
k-polynomial, Lucky χ -polynomial.
AMS Subject Classification
05C15, 05C38, 05C75, 05C85.
Independent Mathematics Researcher, South Africa & Department of Mathematics, CHRIST (Deemed to be a University), Bangalore, India.
*Corresponding author: jacotype@gmail.com; johan.kok@christuniversity.in.
Article History: Received 05 December 2019; Accepted 17 March 2020 ©2020 MJM.
Contents
1 Introduction ....................................... 767
2 Lucky χ -Polynomials .............................. 767
3 Lucky χ -Polynomials of Graphs of Order 5 ....... 769
3.1 Bell partitions ........................ 769
3.2 Category 1: χ (G)= 2 .................. 770
3.3 Category 2: χ (G)= 3 .................. 770
3.4 Category 3: χ (G)= 4 .................. 772
4 Conclusion ........................................ 773
5 Acknowledgments ................................ 773
References ........................................ 773
1. Introduction
For general notation and concepts in graphs see [1,2,7].
Unless stated otherwise, all graphs will be finite and simple
graphs. The set of vertices and the set of edges of a graph
G are denoted by, V (G) and E (G) respectively. The number
of vertices is called the order of G say, n and the number of
edges is called the size of G denoted by, ε (G). If G has order
n ≥ 1 and has no edges (ε (G)= 0) then G is called a null
graph denoted by, N
n
.
For a set of distinct colours C = {c
1
, c
2
, c
3
,..., c
ℓ
} a ver-
tex colouring of a graph G is an assignment ϕ : V (G) → C .
A vertex colouring is said to be a proper vertex colouring of
a graph G if no two distinct adjacent vertices have the same
colour. The cardinality of a minimum set of distinct colours
in a proper vertex colouring of G is called the chromatic
number of G and is denoted χ (G). We call such a colouring
a χ -colouring or a chromatic colouring of G. A chromatic
colouring of G is denoted by ϕ
χ
(G). Generally a graph G
of order n is k-colourable for χ (G) ≤ k ≤ n. The number of
times a colour c
i
is allocated to vertices of a graph G is de-
noted by θ
G
(c
i
) or if the context is clear simply, θ (c
i
).
Generally the set, c( V (G)) ⊆ C . A set {c
i
∈ C : c(v)= c
i
for at least one v ∈ V (G)} is called a colour class of the
colouring of G. If C is the chromatic set it can be agreed that
c(G) means set c( V (G)) hence, c(G) ⇒ C and |c(G)| = |C |.
For the set of vertices X ⊆ V (G), the subgraph induced by
X is denoted by, 〈X 〉. The colouring of 〈X 〉 permitted by
ϕ : V (G) → C is denoted by, c(〈X 〉).
In this paper, Section 2 deals with the introduction to
Lucky χ -polynomials (k = χ (G)). Section 3 presents Lucky
χ -polynomials of all graphs of order 5. Section 4 concludes
the paper and presents problems for further research.
2. Lucky χ -Polynomials
In a proper colouring of G all edges are good i.e. uv ⇔
c(u) = c(v). For any proper colouring ϕ (G) of a graph G
the addition of all good edges, if any, is called the chromatic
completion of G in respect of ϕ (G). The additional edges are