138 research outputs found
Linear colorings of subcubic graphs
A linear coloring of a graph is a proper coloring of the vertices of the
graph so that each pair of color classes induce a union of disjoint paths. In
this paper, we prove that for every connected graph with maximum degree at most
three and every assignment of lists of size four to the vertices of the graph,
there exists a linear coloring such that the color of each vertex belongs to
the list assigned to that vertex and the neighbors of every degree-two vertex
receive different colors, unless the graph is or . This confirms
a conjecture raised by Esperet, Montassier, and Raspaud. Our proof is
constructive and yields a linear-time algorithm to find such a coloring
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
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