46 research outputs found
Some results on multithreshold graphs
Jamison and Sprague defined a graph to be a -threshold graph with
thresholds (strictly increasing) if one can
assign real numbers , called ranks, such that for every
pair of vertices , we have if and only if the inequality
holds for an odd number of indices . When or
, the precise choice of thresholds does not
matter, as a suitable transformation of the ranks transforms a representation
with one choice of thresholds into a representation with any other choice of
thresholds. Jamison asked whether this remained true for or whether
different thresholds define different classes of graphs for such , offering
\C_tt > 13-1, 1, tC_t$, answering Jamison's question. We also
consider some other problems on multithreshold graphs, some of which remain
open.Comment: 6 pages, 1 figur
Graphs with and both large
Given a graph , let denote the smallest size of a set of edges
whose deletion makes triangle-free, and let denote the
largest size of an edge set containing at most one edge from each triangle of
. Erd\H{o}s, Gallai, and Tuza introduced several problems with the unifying
theme that and cannot both be "very large"; the most
well-known such problem is their conjecture that , which was proved by Norin and Sun. We consider three other
problems within this theme (two introduced by Erd\H{o}s, Gallai, and Tuza,
another by Norin and Sun), all of which request an upper bound either on
or on for some
constant , and prove the existence of graphs for which these quantities are
"large".Comment: 6 pages; improved exposition a bit and fixed an issue regarding
integrality from the earlier versio
Complexity of a Disjoint Matching Problem on Bipartite Graphs
We consider the following question: given an -bigraph and a set , does contain two disjoint matchings and such that
saturates and saturates ? When , this
question is solvable by finding an appropriate factor of the graph. In
contrast, we show that when is allowed to be an arbitrary subset of ,
the problem is NP-hard.Comment: 6 pages, 1 figur
Extremal Aspects of the Erd\H{o}s--Gallai--Tuza Conjecture
Erd\H{o}s, Gallai, and Tuza posed the following problem: given an -vertex
graph , let denote the smallest size of a set of edges whose
deletion makes triangle-free, and let denote the largest size
of a set of edges containing at most one edge from each triangle of . Is it
always the case that ? We also consider a
variant on this conjecture: if is the smallest size of an edge set
whose deletion makes bipartite, does the stronger inequality always hold?
By considering the structure of a minimal counterexample to each version of
the conjecture, we obtain two main results. Our first result states that any
minimum counterexample to the original Erd\H{o}s--Gallai--Tuza Conjecture has
"dense edge cuts", and in particular has minimum degree greater than .
This implies that the conjecture holds for all graphs if and only if it holds
for all triangular graphs (graphs where every edge lies in a triangle). Our
second result states that whenever has
no induced subgraph isomorphic to , the graph obtained from the complete
graph by deleting an edge. Thus, the original conjecture also holds for
such graphs.Comment: 5 pages. Updated with journal reference, expanded background, and a
few other minor change
On a Conjecture of Erd\H{o}s, Gallai, and Tuza
Erd\H{o}s, Gallai, and Tuza posed the following problem: given an -vertex
graph , let denote the smallest size of a set of edges whose
deletion makes triangle-free, and let denote the largest size
of a set of edges containing at most one edge from each triangle of . Is it
always the case that ? We have two main
results. We first obtain the upper bound , as a partial result towards the Erd\H{o}s--Gallai--Tuza conjecture.
We also show that always , where is the number
of edges in ; this bound is sharp in several notable cases.Comment: 5 pages, minor revisions: added new details, new conjecture, and
cleaned up notation slightl
Maximal -Edge-Colorable Subgraphs, Vizing's Theorem, and Tuza's Conjecture
We prove that if is a maximal -edge-colorable subgraph of a multigraph
and if , then for all . (When is a simple graph, the set is just the
set of vertices having degree less than in .) This implies Vizing's
Theorem as well as a special case of Tuza's Conjecture on packing and covering
of triangles. A more detailed version of our result also implies Vizing's
Adjacency Lemma for simple graphs.Comment: 11 pages, 1 figure. Fixed some inaccurate references to "Vizing's
Theorem" (the stronger version cited here is in fact due to Ore), cleared up
some muddled results in the section about forests, simplified some notation,
and made other various readability improvement
Favaron's Theorem, k-dependence, and Tuza's Conjecture
A vertex set in a graph is -dependent if has maximum degree
at most , and -dominating if every vertex outside has at least
neighbors in . Favaron proved that if is a -dependent set maximizing
the quantity , then is -dominating. We extend this
result, showing that such sets satisfy a stronger structural property, and we
find a surprising connection between Favaron's theorem and a conjecture of Tuza
regarding packing and covering of triangles.Comment: 12 pages. Strengthened main theorem and simplified its proof by
replacing vertex-orderings with orientation
-cores for -edge-colouring
We extend the edge-coloring notion of core (subgraph induced by the vertices
of maximum degree) to -core (subgraph induced by the vertices with
), and find a sufficient condition for
-edge-coloring. In particular, we show that for any , if
the -core of has multiplicity at most , with its edges of
multiplicity inducing a multiforest, then . This
extends previous work of Ore, Fournier, and Berge and Fournier. A stronger
version of our result (which replaces the multiforest condition with a
vertex-ordering condition) generalizes a theorem of Hoffman and Rodger about
cores of -edge-colourable simple graphs. In fact, our bounds hold not
only for chromatic index, but for the \emph{fan number} of a graph, a parameter
introduced by Scheide and Stiebitz as an upper bound on chromatic index. We are
able to give an exact characterization of the graphs such that
whenever has as its -core.Comment: 15 pages, 2 figures. This version fixes an issue with the definition
of the fan number, and makes several smaller improvement
Environmental Evolutionary Graph Theory
Understanding the influence of an environment on the evolution of its
resident population is a major challenge in evolutionary biology. Great
progress has been made in homogeneous population structures while heterogeneous
structures have received relatively less attention. Here we present a
structured population model where different individuals are best suited to
different regions of their environment. The underlying structure is a graph:
individuals occupy vertices, which are connected by edges. If an individual is
suited for their vertex, they receive an increase in fecundity. This framework
allows attention to be restricted to the spatial arrangement of suitable
habitat. We prove some basic properties of this model and find some
counter-intuitive results. Notably, 1) the arrangement of suitable sites is as
important as their proportion, and, 2) decreasing the proportion of suitable
sites may result in a decrease in the fixation time of an allele
Correlation Clustering and Biclustering with Locally Bounded Errors
We consider a generalized version of the correlation clustering problem,
defined as follows. Given a complete graph whose edges are labeled with
or , we wish to partition the graph into clusters while trying to avoid
errors: edges between clusters or edges within clusters. Classically,
one seeks to minimize the total number of such errors. We introduce a new
framework that allows the objective to be a more general function of the number
of errors at each vertex (for example, we may wish to minimize the number of
errors at the worst vertex) and provide a rounding algorithm which converts
"fractional clusterings" into discrete clusterings while causing only a
constant-factor blowup in the number of errors at each vertex. This rounding
algorithm yields constant-factor approximation algorithms for the discrete
problem under a wide variety of objective functions.Comment: 20 pages, reorganized paper to emphasize the key properties of the
rounding algorithm and the broader class of possible objective function