31,782 research outputs found
On the algorithmic complexity of twelve covering and independence parameters of graphs
The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs
On balanced planar graphs, following W. Thurston
Let be an orientation-preserving branched covering map of
degree , and let be an oriented Jordan curve passing through
the critical values of . Then is an oriented graph
on the sphere. In a group email discussion in Fall 2010, W. Thurston introduced
balanced planar graphs and showed that they combinatorially characterize all
such , where has distinct critical values. We give a
detailed account of this discussion, along with some examples and an appendix
about Hurwitz numbers.Comment: 17 page
Some colouring problems for Paley graphs
The Paley graph Pq, where q≡1(mod4) is a prime power, is the graph with vertices the elements of the finite field Fq and an edge between x and y if and only if x-y is a non-zero square in Fq. This paper gives new results on some colouring problems for Paley graphs and related discussion. © 2005 Elsevier B.V. All rights reserved
Complexes of not -connected graphs
Complexes of (not) connected graphs, hypergraphs and their homology appear in
the construction of knot invariants given by V. Vassiliev. In this paper we
study the complexes of not -connected -hypergraphs on vertices. We
show that the complex of not -connected graphs has the homotopy type of a
wedge of spheres of dimension . This answers one of the
questions raised by Vassiliev in connection with knot invariants. For this case
the -action on the homology of the complex is also determined. For
complexes of not -connected -hypergraphs we provide a formula for the
generating function of the Euler characteristic, and we introduce certain
lattices of graphs that encode their topology. We also present partial results
for some other cases. In particular, we show that the complex of not
-connected graphs is Alexander dual to the complex of partial matchings
of the complete graph. For not -connected graphs we provide a formula
for the generating function of the Euler characteristic
Realizations of self branched coverings of the 2-sphere
For a degree d self branched covering of the 2-sphere, a notable
combinatorial invariant is an integer partition of 2d -- 2, consisting of the
multiplicities of the critical points. A finer invariant is the so called
Hurwitz passport. The realization problem of Hurwitz passports remain largely
open till today. In this article, we introduce two different types of finer
invariants: a bipartite map and an incident matrix. We then settle completely
their realization problem by showing that a map, or a matrix, is realized by a
branched covering if and only if it satisfies a certain balanced condition. A
variant of the bipartite map approach was initiated by W. Thurston. Our results
shed some new lights to the Hurwitz passport problem
Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles
Given a simple graph , a subset of is called a triangle cover if
it intersects each triangle of . Let and denote the
maximum number of pairwise edge-disjoint triangles in and the minimum
cardinality of a triangle cover of , respectively. Tuza conjectured in 1981
that holds for every graph . In this paper, using a
hypergraph approach, we design polynomial-time combinatorial algorithms for
finding small triangle covers. These algorithms imply new sufficient conditions
for Tuza's conjecture on covering and packing triangles. More precisely,
suppose that the set of triangles covers all edges in . We
show that a triangle cover of with cardinality at most can be
found in polynomial time if one of the following conditions is satisfied: (i)
, (ii) , (iii)
.
Keywords: Triangle cover, Triangle packing, Linear 3-uniform hypergraphs,
Combinatorial algorithm
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