753 research outputs found
Acyclic Chromatic Index of Chordless Graphs
An acyclic edge coloring of a graph is a proper edge coloring in which there
are no bichromatic cycles. The acyclic chromatic index of a graph denoted
by , is the minimum positive integer such that has an acyclic
edge coloring with colors. It has been conjectured by Fiam\v{c}\'{\i}k that
for any graph with maximum degree . Linear
arboricity of a graph , denoted by , is the minimum number of linear
forests into which the edges of can be partitioned. A graph is said to be
chordless if no cycle in the graph contains a chord. Every -connected
chordless graph is a minimally -connected graph. It was shown by Basavaraju
and Chandran that if is -degenerate, then . Since
chordless graphs are also -degenerate, we have for any
chordless graph . Machado, de Figueiredo and Trotignon proved that the
chromatic index of a chordless graph is when . They also
obtained a polynomial time algorithm to color a chordless graph optimally. We
improve this result by proving that the acyclic chromatic index of a chordless
graph is , except when and the graph has a cycle, in which
case it is . We also provide the sketch of a polynomial time
algorithm for an optimal acyclic edge coloring of a chordless graph. As a
byproduct, we also prove that , unless
has a cycle with , in which case . To obtain the result on acyclic chromatic
index, we prove a structural result on chordless graphs which is a refinement
of the structure given by Machado, de Figueiredo and Trotignon for this class
of graphs. This might be of independent interest
Equitable partition of graphs into induced forests
An equitable partition of a graph is a partition of the vertex-set of
such that the sizes of any two parts differ by at most one. We show that every
graph with an acyclic coloring with at most colors can be equitably
partitioned into induced forests. We also prove that for any integers
and , any -degenerate graph can be equitably
partitioned into induced forests.
Each of these results implies the existence of a constant such that for
any , any planar graph has an equitable partition into induced
forests. This was conjectured by Wu, Zhang, and Li in 2013.Comment: 4 pages, final versio
Inside-Out Polytopes
We present a common generalization of counting lattice points in rational
polytopes and the enumeration of proper graph colorings, nowhere-zero flows on
graphs, magic squares and graphs, antimagic squares and graphs, compositions of
an integer whose parts are partially distinct, and generalized latin squares.
Our method is to generalize Ehrhart's theory of lattice-point counting to a
convex polytope dissected by a hyperplane arrangement. We particularly develop
the applications to graph and signed-graph coloring, compositions of an
integer, and antimagic labellings.Comment: 24 pages, 3 figures; to appear in Adv. Mat
Boxicity and topological invariants
The boxicity of a graph is the smallest integer for which there
exist interval graphs , , such that . In the first part of this note, we prove that every graph on
edges has boxicity , which is asymptotically best
possible. We use this result to study the connection between the boxicity of
graphs and their Colin de Verdi\`ere invariant, which share many similarities.
Known results concerning the two parameters suggest that for any graph , the
boxicity of is at most the Colin de Verdi\`ere invariant of , denoted by
. We observe that every graph has boxicity , while there are graphs with boxicity . In the second part of this note, we focus on graphs embeddable on a
surface of Euler genus . We prove that these graphs have boxicity
, while some of these graphs have boxicity . This improves the previously best known upper and lower bounds.
These results directly imply a nearly optimal bound on the dimension of the
adjacency poset of graphs on surfaces.Comment: 6 page
- …