20,332 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
On generalized Ramsey numbers in the non-integral regime
A -coloring of a graph is an edge-coloring of such that every
-clique receives at least colors. In 1975, Erd\H{o}s and Shelah
introduced the generalized Ramsey number which is the minimum number
of colors needed in a -coloring of . In 1997, Erd\H{o}s and
Gy\'arf\'as showed that is at most a constant times
. Very recently the first author, Dudek,
and English improved this bound by a factor of for all , and they ask if this
improvement could hold for a wider range of .
We answer this in the affirmative for the entire non-integral regime, that
is, for all integers with not divisible by .
Furthermore, we provide a simultaneous three-way generalization as follows:
where -clique is replaced by any fixed graph (with not
divisible by ); to list coloring; and to -uniform
hypergraphs. Our results are a new application of the Forbidden Submatching
Method of the second and fourth authors.Comment: 9 pages; new version extends results from sublinear regime to entire
non-integral regime; new co-author adde
Acyclic edge-coloring using entropy compression
An edge-coloring of a graph G is acyclic if it is a proper edge-coloring of G
and every cycle contains at least three colors. We prove that every graph with
maximum degree Delta has an acyclic edge-coloring with at most 4 Delta - 4
colors, improving the previous bound of 9.62 (Delta - 1). Our bound results
from the analysis of a very simple randomised procedure using the so-called
entropy compression method. We show that the expected running time of the
procedure is O(mn Delta^2 log Delta), where n and m are the number of vertices
and edges of G. Such a randomised procedure running in expected polynomial time
was only known to exist in the case where at least 16 Delta colors were
available. Our aim here is to make a pedagogic tutorial on how to use these
ideas to analyse a broad range of graph coloring problems. As an application,
also show that every graph with maximum degree Delta has a star coloring with 2
sqrt(2) Delta^{3/2} + Delta colors.Comment: 13 pages, revised versio
Almost-rainbow edge-colorings of some small subgraphs
Let be the minimum number of colors necessary to color the edges
of so that every is at least -colored. We improve current bounds
on the {7/4}n-3{5/6}n+1\leq
f(n,4,5)n\not\equiv 1 \pmod 3f(n,4,5)\leq n-1G=K_{n,n}GC_4\subseteq G$ is colored by at least three
colors. This improves the best known upper bound of M. Axenovich, Z. F\"uredi,
and D. Mubayi.Comment: 13 page
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