593 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
A lower bound on the order of the largest induced forest in planar graphs with high girth
We give here new upper bounds on the size of a smallest feedback vertex set
in planar graphs with high girth. In particular, we prove that a planar graph
with girth and size has a feedback vertex set of size at most
, improving the trivial bound of . We also prove
that every -connected graph with maximum degree and order has a
feedback vertex set of size at most .Comment: 12 pages, 6 figures. arXiv admin note: text overlap with
arXiv:1409.134
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
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