SOME GRAPHS IN $C_f2$ BASED ON $f$-COLORING

Adiwijaya Adiwijaya,A.N.M Salman,O Serra,D Suprijanto,E.T Baskoro

doi:10.12732/ijpam.v102i2.3

Abstract

Let G = (V,E) be a graph and f : V → Z + a positive integer be a function. An f-coloring of G is a coloring of the edges such that every vertex v ∈ V is incident to at most f(v) edges of the same color. The minimum number of colors of an f-coloring of G is the f-chromatic index � ' (G) of G. Based on the f-chromatic index, a graph G can be either in class Cf1, if � ' (G) = �f(G), or in class Cf2, if � ' (G) = �f(G)+1, wheref(G) = maxx∈V ⌈d(v)/f(v)⌉. In this paper, we give some sufficient conditions for a graph to bein Cf2. One of the results is a generalization of a theorem by Zhang et al. (2008). Moreover, we show that, when f is constant and a divisor of (n−1), a maximal subgraph of the complete graph Kn which is in class Cf1 has precisely n � − �f(Kn)/2 edges.

Highlights

Let G = (V, E) be a finite and simple graph and let f be a function from V to a positive integer set
If k and n are odd integers with n ≥ 3, f (v) = k for all v ∈ V (Kn), and k divides n − 1, the complete graph Kn is in Cf 2
In Theorem 5, we provide an edge-reduction of a complete graph which is in Cf 2 in order to get a maximal subgraph of Kn which is in Cf 1

Summary

Introduction

In case f (v) = 1 for every v ∈ V , an f -coloring is just a proper edge-coloring of the graph. Zhang and Liu [6] gave the following classification of complete graphs based on f -colorings. If k and n are odd integers with n ≥ 3, f (v) = k for all v ∈ V (Kn), and k divides n − 1, the complete graph Kn is in Cf 2.

Results

Conclusion