45 research outputs found
Planar graphs are acyclically edge -colorable
An edge coloring of a graph is to color all the edges in the graph such
that adjacent edges receive different colors. It is acyclic if each cycle in
the graph receives at least three colors. Fiam{\v{c}}ik (1978) and Alon,
Sudakov and Zaks (2001) conjectured that every simple graph with maximum degree
is acyclically edge -colorable -- the well-known acyclic
edge coloring conjecture (AECC). Despite many major breakthroughs and minor
improvements, the conjecture remains open even for planar graphs. In this
paper, we prove that planar graphs are acyclically edge -colorable. Our proof has two main steps: Using discharging methods, we
first show that every non-trivial planar graph must have one of the eight
groups of well characterized local structures; and then acyclically edge color
the graph using no more than colors by an induction on the number
of edges.Comment: Full version with 120 page
Graph Partitioning With Input Restrictions
In this thesis we study the computational complexity of a number of graph
partitioning problems under a variety of input restrictions. Predominantly,
we research problems related to Colouring in the case where the input
is limited to hereditary graph classes, graphs of bounded diameter or some
combination of the two.
In Chapter 2 we demonstrate the dramatic eect that restricting our
input to hereditary graph classes can have on the complexity of a decision
problem. To do this, we show extreme jumps in the complexity of three
problems related to graph colouring between the class of all graphs and every
other hereditary graph class.
We then consider the problems Colouring and k-Colouring for Hfree graphs of bounded diameter in Chapter 3. A graph class is said to be
H-free for some graph H if it contains no induced subgraph isomorphic to
H. Similarly, G is said to be H-free for some set of graphs H, if it does not
contain any graph in H as an induced subgraph. Here, the set H consists
usually of a single cycle or tree but may also contain a number of cycles, for
example we give results for graphs of bounded diameter and girth.
Chapter 4 is dedicated to three variants of the Colouring problem,
Acyclic Colouring, Star Colouring, and Injective Colouring.
We give complete or almost complete dichotomies for each of these decision
problems restricted to H-free graphs.
In Chapter 5 we study these problems, along with three further variants of
3-Colouring, Independent Odd Cycle Transversal, Independent
Feedback Vertex Set and Near-Bipartiteness, for H-free graphs of
bounded diameter.
Finally, Chapter 6 deals with a dierent variety of problems. We study
the problems Disjoint Paths and Disjoint Connected Subgraphs for
H-free graphs
Equitable partition of planar graphs
An equitable -partition of a graph is a collection of induced
subgraphs of such that
is a partition of and
for all . We prove that every planar graph admits an equitable
-partition into -degenerate graphs, an equitable -partition into
-degenerate graphs, and an equitable -partition into two forests and one
graph.Comment: 12 pages; revised; accepted to Discrete Mat