54 research outputs found
Acyclic Subgraphs of Planar Digraphs
An acyclic set in a digraph is a set of vertices that induces an acyclic
subgraph. In 2011, Harutyunyan conjectured that every planar digraph on
vertices without directed 2-cycles possesses an acyclic set of size at least
. We prove this conjecture for digraphs where every directed cycle has
length at least 8. More generally, if is the length of the shortest
directed cycle, we show that there exists an acyclic set of size at least .Comment: 9 page
Structural properties of 1-planar graphs and an application to acyclic edge coloring
A graph is called 1-planar if it can be drawn on the plane so that each edge
is crossed by at most one other edge. In this paper, we establish a local
property of 1-planar graphs which describes the structure in the neighborhood
of small vertices (i.e. vertices of degree no more than seven). Meanwhile, some
new classes of light graphs in 1-planar graphs with the bounded degree are
found. Therefore, two open problems presented by Fabrici and Madaras [The
structure of 1-planar graphs, Discrete Mathematics, 307, (2007), 854-865] are
solved. Furthermore, we prove that each 1-planar graph with maximum degree
is acyclically edge -choosable where
.Comment: Please cite this published article as: X. Zhang, G. Liu, J.-L. Wu.
Structural properties of 1-planar graphs and an application to acyclic edge
coloring. Scientia Sinica Mathematica, 2010, 40, 1025--103
Acyclic 4-choosability of planar graphs without 4-cycles
summary:A proper vertex coloring of a graph is acyclic if there is no bicolored cycle in . In other words, each cycle of must be colored with at least three colors. Given a list assignment , if there exists an acyclic coloring of such that for all , then we say that is acyclically -colorable. If is acyclically -colorable for any list assignment with for all , then is acyclically -choosable. In 2006, Montassier, Raspaud and Wang conjectured that every planar graph without 4-cycles is acyclically 4-choosable. However, this has been as yet verified only for some restricted classes of planar graphs. In this paper, we prove that every planar graph with neither 4-cycles nor intersecting -cycles for each is acyclically 4-choosable
Acyclic edge coloring of subcubic graphs
AbstractAn acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and it is denoted by aβ²(G). From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using five colors. This result is tight since there are 3-regular graphs which require five colors. In this paper we prove that any non-regular connected graph of maximum degree 3 is acyclically edge colorable using at most four colors. This result is tight since all edge maximal non-regular connected graphs of maximum degree 3 require four colors
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