L(t, 1)-Colouring of Graphs

TL;DR

This paper introduces L(t, 1)-colouring, a new graph colouring concept inspired by T-colouring and L(p, q)-colouring, with applications in channel assignment problems, and studies its properties and bounds.

Contribution

It defines the L(t, 1)-colouring of graphs, explores its properties, and establishes upper bounds for simple connected graphs, advancing graph colouring theory.

Findings

Defined L(t, 1)-colouring and its parameters.

Derived upper bounds for specific graph classes.

Analyzed properties related to frequency assignment applications.

Abstract

One of the most famous applications of Graph Theory is in the field of Channel Assignment Problems. There are varieties of graph colouring concepts that are used for different requirements of frequency assignments in communication channels. We introduce here L(t, 1)-colouring of graphs. This has its foundation in T-colouring and L(p, q)-colouring. For a given finite set T including zero, an L(t, 1)-colouring of a graph G is an assignment of non-negative integers to the vertices of G such that the difference between the colours of adjacent vertices must not belong to the set T and the colours of vertices that are at distance two must be distinct. The variable t in L(t, 1) denotes the elements of the set T. For a graph G, the L(t, 1)-span of G is the minimum of the highest colour used to colour the vertices of a graph out of all the possible L(t, 1)-colourings. It is denoted by $\lambda_{t,1} (G)$. We study some properties of L(t, 1)-colouring. We also find upper bounds of $\lambda_{t,1} (G)$ of selected simple connected graphs.