39 research outputs found
Some Results on incidence coloring, star arboricity and domination number
Two inequalities bridging the three isolated graph invariants, incidence
chromatic number, star arboricity and domination number, were established.
Consequently, we deduced an upper bound and a lower bound of the incidence
chromatic number for all graphs. Using these bounds, we further reduced the
upper bound of the incidence chromatic number of planar graphs and showed that
cubic graphs with orders not divisible by four are not 4-incidence colorable.
The incidence chromatic numbers of Cartesian product, join and union of graphs
were also determined.Comment: 8 page
The Incidence Chromatic Number of Toroidal Grids
An incidence in a graph is a pair with and , such that and are incident. Two incidences and
are adjacent if , or , or the edge equals or . The
incidence chromatic number of is the smallest for which there exists a
mapping from the set of incidences of to a set of colors that assigns
distinct colors to adjacent incidences. In this paper, we prove that the
incidence chromatic number of the toroidal grid equals 5
when and 6 otherwise.Comment: 16 page
Graph Treewidth and Geometric Thickness Parameters
Consider a drawing of a graph in the plane such that crossing edges are
coloured differently. The minimum number of colours, taken over all drawings of
, is the classical graph parameter "thickness". By restricting the edges to
be straight, we obtain the "geometric thickness". By further restricting the
vertices to be in convex position, we obtain the "book thickness". This paper
studies the relationship between these parameters and treewidth.
Our first main result states that for graphs of treewidth , the maximum
thickness and the maximum geometric thickness both equal .
This says that the lower bound for thickness can be matched by an upper bound,
even in the more restrictive geometric setting. Our second main result states
that for graphs of treewidth , the maximum book thickness equals if and equals if . This refutes a conjecture of Ganley and
Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved
for outerthickness, arboricity, and star-arboricity.Comment: A preliminary version of this paper appeared in the "Proceedings of
the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in
Computer Science 3843:129-140, Springer, 2006. The full version was published
in Discrete & Computational Geometry 37(4):641-670, 2007. That version
contained a false conjecture, which is corrected on page 26 of this versio
On globally sparse Ramsey graphs
We say that a graph has the Ramsey property w.r.t.\ some graph and
some integer , or is -Ramsey for short, if any -coloring
of the edges of contains a monochromatic copy of . R{\"o}dl and
Ruci{\'n}ski asked how globally sparse -Ramsey graphs can possibly
be, where the density of is measured by the subgraph with
the highest average degree. So far, this so-called Ramsey density is known only
for cliques and some trivial graphs . In this work we determine the Ramsey
density up to some small error terms for several cases when is a complete
bipartite graph, a cycle or a path, and colors are available
Planar Ramsey graphs
We say that a graph is planar unavoidable if there is a planar graph
such that any red/blue coloring of the edges of contains a monochromatic
copy of , otherwise we say that is planar avoidable. I.e., is planar
unavoidable if there is a Ramsey graph for that is planar. It follows from
the Four-Color Theorem and a result of Gon\c{c}alves that if a graph is planar
unavoidable then it is bipartite and outerplanar. We prove that the cycle on
vertices and any path are planar unavoidable. In addition, we prove that
all trees of radius at most are planar unavoidable and there are trees of
radius that are planar avoidable. We also address the planar unavoidable
notion in more than two colors
The acircuitic directed star arboricity of subcubic graphs is at most four
AbstractA directed star forest is a forest all of whose components are stars with arcs emanating from the center to the leaves. The acircuitic directed star arboricity of an oriented graph G (that is a digraph with no opposite arcs) is the minimum number of arc-disjoint directed star forests whose union covers all arcs of G and such that the union of any two such forests is acircuitic. We show that every subcubic graph has acircuitic directed star arboricity at most four