14,358 research outputs found
On star edge colorings of bipartite and subcubic graphs
A star edge coloring of a graph is a proper edge coloring with no -colored
path or cycle of length four. The star chromatic index of
is the minimum number for which has a star edge coloring with
colors. We prove upper bounds for the star chromatic index of complete
bipartite graphs; in particular we obtain tight upper bounds for the case when
one part has size at most . We also consider bipartite graphs where all
vertices in one part have maximum degree and all vertices in the other part
has maximum degree . Let be an integer (), we prove that if
then ; and if , then ; both upper bounds are sharp.
Finally, we consider the well-known conjecture that subcubic graphs have star
chromatic index at most ; in particular we settle this conjecture for cubic
Halin graphs.Comment: 18 page
Edge and total colourings of graphs
Die vorliegenden Arbeit enthält Ergebnisse zu Kanten- und Totalfärbungen von Graphen sowie verschiedenen Variationen dieser Färbungen. Eine Kantenfärbung eines Graphen G ist eine Zuordnung von Farben zu den Kanten von G, so dass adjazente Kanten unterschiedliche Farben erhalten. Eine Totalfärbung ist eine Färbung der Knoten und Kanten von G, so dass adjazente Knoten, adjazente Kanten sowie ein Knoten und eine inzidente Kante jeweils unterschiedlich gefärbt werden. Der chromatische Index bzw. die totalchromatische Zahl von G bezeichnen die kleinste Anzahl von Farben, mit denen G kantenfärbbar bzw. totalfärbbar ist. In dieser Arbeit wird unter anderem die totalchromatische Zahl zirkulanter Graphen mit Maximalgrad 3 bestimmt sowie ein Algorithmus entwickelt, der alle planaren kritischen Graphen der Kantenfärbung mit bis zu 12 Knoten konstruiert und darstellt. Das Konzept der Kreisfärbung von Graphen wird von Knoten- auf Kanten- und Totalfärbung übertragen; Eigenschaften des kreischromatischen Index und der kreistotalchromatischen Zahl werden bewiesen und exakte Werte für einige Graphenklassen ermittelt. Die listenchromatische Vermutung wird für outerplanare Graphen mit Maximalgrad >4 bewiesen. Die Konzepte der (a,b)- und (a,b,r)-Listen- färbung werden von Knotenfärbung auf Kantenfärbung übertragen; es werden Eigenschaften dieser Färbungen und Ergebnisse für einzelne Graphenklassen hergeleitet.This thesis contains results for edge and total colourings as well as for some variations of these colourings. An edge colouring of a graph G is an assignment of colours to the edges of G such that adjacent edges are coloured differently. A total colouring is a colouring of the vertices and edges of G such that adjacent vertices, adjacent edges as well as a vertex and an incident edge are coloured differently. The chromatic index or the total chromatic number of G denote the minimum number of colours such that G admits an edge colouring or a total colouring, respectively. Results in this thesis are - among others - the total chromatic number of circulant graphs with maximum degree 3 and an algorithm to construct and draw all planar critical graphs with at most 12 vertices. The concept of circular colourings is transferred from vertex to edge and total colourings. Properties of the circular chromatic index and the circular total chromatic number are proven and exact values are determined for some classes of graphs. The list chromatic conjecture is confirmed for outerplanar graphs with maximum degree >4; the concepts of (a,b)- and (a,b,r)-list colourings are transferred from vertex to edge colouring and properties of these colourings as well as results for special classes of graphs are given
Local Irregularity Conjecture vs. cacti
A graph is locally irregular if the degrees of the end-vertices of every edge
are distinct. An edge coloring of a graph G is locally irregular if every color
induces a locally irregular subgraph of G. A colorable graph G is any graph
which admits a locally irregular edge coloring. The locally irregular chromatic
index X'irr(G) of a colorable graph G is the smallest number of colors required
by a locally irregular edge coloring of G. The Local Irregularity Conjecture
claims that all colorable graphs require at most 3 colors for locally irregular
edge coloring. Recently, it has been observed that the conjecture does not hold
for the bow-tie graph B [7]. Cacti are important class of graphs for this
conjecture since B and all non-colorable graphs are cacti. In this paper we
show that for every colorable cactus graph G != B it holds that X'irr(G) <= 3.
This makes us to believe that B is the only colorable graph with X'irr(B) > 3,
and consequently that B is the only counterexample to the Local Irregularity
Conjecture.Comment: 27 pages, 7 figure
Local Graph Coloring and Index Coding
We present a novel upper bound for the optimal index coding rate. Our bound
uses a graph theoretic quantity called the local chromatic number. We show how
a good local coloring can be used to create a good index code. The local
coloring is used as an alignment guide to assign index coding vectors from a
general position MDS code. We further show that a natural LP relaxation yields
an even stronger index code. Our bounds provably outperform the state of the
art on index coding but at most by a constant factor.Comment: 14 Pages, 3 Figures; A conference version submitted to ISIT 2013;
typos correcte
Strong chromatic index of sparse graphs
A coloring of the edges of a graph is strong if each color class is an
induced matching of . The strong chromatic index of , denoted by
, is the least number of colors in a strong edge coloring
of . In this note we prove that for every -degenerate graph . This confirms the strong
version of conjecture stated recently by Chang and Narayanan [3]. Our approach
allows also to improve the upper bound from [3] for chordless graphs. We get
that for any chordless graph . Both
bounds remain valid for the list version of the strong edge coloring of these
graphs
- …