219 research outputs found
A DSATUR-based algorithm for the Equitable Coloring Problem
This paper describes a new exact algorithm for the Equitable Coloring Problem, a coloring problem where the sizes of two arbitrary color classes differ in at most one unit. Based on the well known DSatur algorithm for the classic Coloring Problem, a pruning criterion arising from equity constraints is proposed and analyzed. The good performance of the algorithm is shown through computational experiments over random and benchmark instances.Fil: Méndez-Díaz, Isabel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Nasini, Graciela Leonor. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Severin, Daniel Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentin
On perfect packings in dense graphs
We say that a graph G has a perfect H-packing if there exists a set of
vertex-disjoint copies of H which cover all the vertices in G. We consider
various problems concerning perfect H-packings: Given positive integers n, r,
D, we characterise the edge density threshold that ensures a perfect
K_r-packing in any graph G on n vertices and with minimum degree at least D. We
also give two conjectures concerning degree sequence conditions which force a
graph to contain a perfect H-packing. Other related embedding problems are also
considered. Indeed, we give a structural result concerning K_r-free graphs that
satisfy a certain degree sequence condition.Comment: 18 pages, 1 figure. Electronic Journal of Combinatorics 20(1) (2013)
#P57. This version contains an open problem sectio
Ore-degree threshold for the square of a Hamiltonian cycle
A classic theorem of Dirac from 1952 states that every graph with minimum
degree at least n/2 contains a Hamiltonian cycle. In 1963, P\'osa conjectured
that every graph with minimum degree at least 2n/3 contains the square of a
Hamiltonian cycle. In 1960, Ore relaxed the degree condition in the Dirac's
theorem by proving that every graph with for every contains a Hamiltonian cycle. Recently, Ch\^au proved an Ore-type
version of P\'osa's conjecture for graphs on vertices using the
regularity--blow-up method; consequently the is very large (involving a
tower function). Here we present another proof that avoids the use of the
regularity lemma. Aside from the fact that our proof holds for much smaller
, we believe that our method of proof will be of independent interest.Comment: 24 pages, 1 figure. In addition to some fixed typos, this updated
version contains a simplified "connecting lemma" in Section 3.
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