148 research outputs found
On the neighbour sum distinguishing index of planar graphs
Let be a proper edge colouring of a graph with integers
. Then , while by Vizing's theorem, no more than
is necessary for constructing such . On the course of
investigating irregularities in graphs, it has been moreover conjectured that
only slightly larger , i.e., enables enforcing additional
strong feature of , namely that it attributes distinct sums of incident
colours to adjacent vertices in if only this graph has no isolated edges
and is not isomorphic to . We prove the conjecture is valid for planar
graphs of sufficiently large maximum degree. In fact even stronger statement
holds, as the necessary number of colours stemming from the result of Vizing is
proved to be sufficient for this family of graphs. Specifically, our main
result states that every planar graph of maximum degree at least which
contains no isolated edges admits a proper edge colouring
such that for every edge of .Comment: 22 page
Locally identifying colourings for graphs with given maximum degree
A proper vertex-colouring of a graph G is said to be locally identifying if
for any pair u,v of adjacent vertices with distinct closed neighbourhoods, the
sets of colours in the closed neighbourhoods of u and v are different. We show
that any graph G has a locally identifying colouring with
colours, where is the maximum degree of G, answering in a positive way
a question asked by Esperet et al. We also provide similar results for locally
identifying colourings which have the property that the colours in the
neighbourhood of each vertex are all different and apply our method to the
class of chordal graphs
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Aspects of graph colouring
The four-colour conjecture of 1852, and the total colouring conjecture of 1965, have sparked off many new concepts and conjectures. In this thesis we investigate many of the outstanding conjectures, establishing various related results, and present many conjectures of our own. We give a brief historical introduction (Chapter 1) and establish some notation, terminology and techniques (Chapter 2). Next, in Chapter 3, we examine the use of latin squares to represent edge and total colourings. In Chapters 4 - 6 we deal with vertex, edge and total colourings respectively. Various ways of measuring different aspects of graphs are presented, in particular, the ‘colouring difference’ between two edge-colourings of a graph (Chapter 5) and the ‘beta parameter’ (defined in Chapter 2 and used in Chapters 3 and 6); this is a measure of how far from a type 1 graph a type 2 graph can be. In Chapter 6 we derive an upper bound for the beta value of any near type 1 graph and give the exact results for all Kn. The number of ways of colouring Kn and Kn,,n are also quantified. Chapter 6 also examines Hilton’s concept of conformability. It is shown that every graph with at least A spines is conformable, and an extension to the concept, which we call G*-conformability, is introduced. We then give new necessary conditions for a cubic graph to be type 1 in relation to G*-conformability. Various methods of manipulating graphs are considered and we present: a method to compatibly triangulate a graph G-e; a method of introducing a fourth colour thus allowing a sequence of Kempe interchanges from any edge 3-colouring of a cubic graph to any other; and a method to re-colour a near type 1 graph within a certain bound on beta. We end this thesis with a brief discussion on possible practical uses for colouring graphs. A list of the main results and conjectures is given at the end of each chapter, but a short list of the principle theorems proven is given below
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