26 research outputs found
Vertex arboricity of triangle-free graphs
Master's Project (M.S.) University of Alaska Fairbanks, 2016The vertex arboricity of a graph is the minimum number of colors needed to color the vertices so that the subgraph induced by each color class is a forest. In other words, the vertex arboricity of a graph is the fewest number of colors required in order to color a graph such that every cycle has at least two colors. Although not standard, we will refer to vertex arboricity simply as arboricity. In this paper, we discuss properties of chromatic number and k-defective chromatic number and how those properties relate to the arboricity of trianglefree graphs. In particular, we find bounds on the minimum order of a graph having arboricity three. Equivalently, we consider the largest possible vertex arboricity of triangle-free graphs of fixed order
Composing dynamic programming tree-decomposition-based algorithms
Given two integers and as well as graph classes
, the problems
,
, and
ask, given graph
as input, whether , , respectively can be partitioned
into sets such that, for each between and
, , , respectively. Moreover in , we request that the number of edges with
endpoints in different sets of the partition is bounded by . We show that if
there exist dynamic programming tree-decomposition-based algorithms for
recognizing the graph classes , for each , then we can
constructively create a dynamic programming tree-decomposition-based algorithms
for ,
, and
. We show that, in
some known cases, the obtained running times are comparable to those of the
best know algorithms
Three ways to cover a graph
We consider the problem of covering an input graph with graphs from a
fixed covering class . The classical covering number of with respect to
is the minimum number of graphs from needed to cover the edges of
without covering non-edges of . We introduce a unifying notion of three
covering parameters with respect to , two of which are novel concepts only
considered in special cases before: the local and the folded covering number.
Each parameter measures "how far'' is from in a different way. Whereas
the folded covering number has been investigated thoroughly for some covering
classes, e.g., interval graphs and planar graphs, the local covering number has
received little attention.
We provide new bounds on each covering number with respect to the following
covering classes: linear forests, star forests, caterpillar forests, and
interval graphs. The classical graph parameters that result this way are
interval number, track number, linear arboricity, star arboricity, and
caterpillar arboricity. As input graphs we consider graphs of bounded
degeneracy, bounded degree, bounded tree-width or bounded simple tree-width, as
well as outerplanar, planar bipartite, and planar graphs. For several pairs of
an input class and a covering class we determine exactly the maximum ordinary,
local, and folded covering number of an input graph with respect to that
covering class.Comment: 20 pages, 4 figure
Injective edge coloring of graphs
Three edges and in a graph are consecutive if they form a path (in this order) or a cycle of lengths three. An injective edge coloring of a graph is a coloring of the edges of such that if and are consecutive edges in , then . The injective edge coloring number is the minimum number of colors permitted in such a coloring. In this paper, exact values of for several classes of graphs are obtained, upper and lower bounds for are introduced and it is proven that checking whether is NP-complete.in publicatio