61 research outputs found
Variations on the Roy-Gallai theorem
Abstract.: A generalization of the Roy-Gallai Theorem on the chromatic number of a graph is derived which is also an extension of several other results of Berge and of Li. A simple inductive proof is given which provides a direct way of deriving the Theorem of Li. We also show that some classical results valid for optimal colorings cannot be transposed to suboptimal colorings. We finally investigate some elementary properties which are also valid in suboptimal coloring
Berge's conjecture on directed path partitions—a survey
AbstractBerge's conjecture from 1982 on path partitions in directed graphs generalizes and extends Dilworth's theorem and the Greene–Kleitman theorem which are well known for partially ordered sets. The conjecture relates path partitions to a collection of k independent sets, for each k⩾1. The conjecture is still open and intriguing for all k>1.11Only recently it was proved Berger and Ben-Arroyo Hartman [56] for k=2 (added in proof). In this paper, we will survey partial results on the conjecture, look into different proof techniques for these results, and relate the conjecture to other theorems, conjectures and open problems of Berge and other mathematicians
Parameter identifiability in a class of random graph mixture models
We prove identifiability of parameters for a broad class of random graph
mixture models. These models are characterized by a partition of the set of
graph nodes into latent (unobservable) groups. The connectivities between nodes
are independent random variables when conditioned on the groups of the nodes
being connected. In the binary random graph case, in which edges are either
present or absent, these models are known as stochastic blockmodels and have
been widely used in the social sciences and, more recently, in biology. Their
generalizations to weighted random graphs, either in parametric or
non-parametric form, are also of interest in many areas. Despite a broad range
of applications, the parameter identifiability issue for such models is
involved, and previously has only been touched upon in the literature. We give
here a thorough investigation of this problem. Our work also has consequences
for parameter estimation. In particular, the estimation procedure proposed by
Frank and Harary for binary affiliation models is revisited in this article
Hypohamiltonian and almost hypohamiltonian graphs
This Dissertation is structured as follows. In Chapter 1, we give a short historical overview and define fundamental concepts. Chapter 2 contains a clear narrative of the progress made towards finding the smallest planar hypohamiltonian graph, with all of the necessary theoretical tools and techniques--especially Grinberg's Criterion. Consequences of this progress are distributed over all sections and form the leitmotif of this Dissertation. Chapter 2 also treats girth restrictions and hypohamiltonian graphs in the context of crossing numbers. Chapter 3 is a thorough discussion of the newly introduced almost hypohamiltonian graphs and their connection to hypohamiltonian graphs. Once more, the planar case plays an exceptional role. At the end of the chapter, we study almost hypotraceable graphs and Gallai's problem on longest paths. The latter leads to Chapter 4, wherein the connection between hypohamiltonicity and various problems related to longest paths and longest cycles are presented. Chapter 5 introduces and studies non-hamiltonian graphs in which every vertex-deleted subgraph is traceable, a class encompassing hypohamiltonian and hypotraceable graphs. We end with an outlook in Chapter 6, where we present a selection of open problems enriched with comments and partial results
Turán Number of an Induced Complete Bipartite Graph Plus an Odd Cycle
Let k ⩾ 2 be an integer. We show that if s = 2 and t ⩾ 2, or s = t = 3, then the maximum possible number of edges in a C2k+1-free graph containing no induced copy of Ks,t is asymptotically equal to (t − s + 1)1/s(n/2)2−1/s except when k = s = t = 2. This strengthens a result of Allen, Keevash, Sudakov and Verstraëte [1], and answers a question of Loh, Tait, Timmons and Zhou [14]. Copyright © Cambridge University Press 201
Tropical Dominating Sets in Vertex-Coloured Graphs
Given a vertex-coloured graph, a dominating set is said to be tropical if
every colour of the graph appears at least once in the set. Here, we study
minimum tropical dominating sets from structural and algorithmic points of
view. First, we prove that the tropical dominating set problem is NP-complete
even when restricted to a simple path. Then, we establish upper bounds related
to various parameters of the graph such as minimum degree and number of edges.
We also give upper bounds for random graphs. Last, we give approximability and
inapproximability results for general and restricted classes of graphs, and
establish a FPT algorithm for interval graphs.Comment: 19 pages, 4 figure
Rank-Dependent Measures of Bi-Polarization and Marginal Tax Reforms
In this paper, we investigate a dual class of bi-polarization indices, namely rank-dependent bi-polarization indices. We show that these indices may be characterized with the generalized positional transfer sensitivity property. We find necessary and sufficient conditions in order to identify bi-polarization-reducing marginal tax reforms. Precisely, we propose inverse positional dominance criteria based on the comparison of bi-polarization concentration curves. An illustration is presented using the Jordanian Household Expenditure and Income Survey 2002/2003.
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