25 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
Acyclic list edge coloring of outerplanar graphs
AbstractAn acyclic list edge coloring of a graph G is a proper list edge coloring such that no bichromatic cycles are produced. In this paper, we prove that an outerplanar graph G with maximum degree Δ≥5 has the acyclic list edge chromatic number equal to Δ
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