48 research outputs found
Acyclic edge coloring of graphs
An {\em acyclic edge coloring} of a graph is a proper edge coloring such
that the subgraph induced by any two color classes is a linear forest (an
acyclic graph with maximum degree at most two). The {\em acyclic chromatic
index} \chiup_{a}'(G) of a graph is the least number of colors needed in
an acyclic edge coloring of . Fiam\v{c}\'{i}k (1978) conjectured that
\chiup_{a}'(G) \leq \Delta(G) + 2, where is the maximum degree of
. This conjecture is well known as Acyclic Edge Coloring Conjecture (AECC).
A graph with maximum degree at most is {\em
-deletion-minimal} if \chiup_{a}'(G) > \kappa and \chiup_{a}'(H)
\leq \kappa for every proper subgraph of . The purpose of this paper is
to provide many structural lemmas on -deletion-minimal graphs. By using
the structural lemmas, we firstly prove that AECC is true for the graphs with
maximum average degree less than four (\autoref{NMAD4}). We secondly prove that
AECC is true for the planar graphs without triangles adjacent to cycles of
length at most four, with an additional condition that every -cycle has at
most three edges contained in triangles (\autoref{NoAdjacent}), from which we
can conclude some known results as corollaries. We thirdly prove that every
planar graph without intersecting triangles satisfies \chiup_{a}'(G) \leq
\Delta(G) + 3 (\autoref{NoIntersect}). Finally, we consider one extreme case
and prove it: if is a graph with and all the
-vertices are independent, then \chiup_{a}'(G) = \Delta(G). We hope
the structural lemmas will shed some light on the acyclic edge coloring
problems.Comment: 19 page
Unsolved Problems in Spectral Graph Theory
Spectral graph theory is a captivating area of graph theory that employs the
eigenvalues and eigenvectors of matrices associated with graphs to study them.
In this paper, we present a collection of topics in spectral graph theory,
covering a range of open problems and conjectures. Our focus is primarily on
the adjacency matrix of graphs, and for each topic, we provide a brief
historical overview.Comment: v3, 30 pages, 1 figure, include comments from Clive Elphick, Xiaofeng
Gu, William Linz, and Dragan Stevanovi\'c, respectively. Thanks! This paper
will be published in Operations Research Transaction
Measurements of edge uncolourability in cubic graphs
Philosophiae Doctor - PhDThe history of the pursuit of uncolourable cubic graphs dates back more than a century.
This pursuit has evolved from the slow discovery of individual uncolourable
cubic graphs such as the famous Petersen graph and the Blanusa snarks, to discovering
in nite classes of uncolourable cubic graphs such as the Louphekine and
Goldberg snarks, to investigating parameters which measure the uncolourability of
cubic graphs. These parameters include resistance, oddness and weak oddness,
ow
resistance, among others. In this thesis, we consider current ideas and problems regarding
the uncolourability of cubic graphs, centering around these parameters. We
introduce new ideas regarding the structural complexity of these graphs in question.
In particular, we consider their 3-critical subgraphs, speci cally in relation to resistance.
We further introduce new parameters which measure the uncolourability of
cubic graphs, speci cally relating to their 3-critical subgraphs and various types of
cubic graph reductions. This is also done with a view to identifying further problems
of interest. This thesis also presents solutions and partial solutions to long-standing
open conjectures relating in particular to oddness, weak oddness and resistance