24 research outputs found
Nonextendible Latin Cuboids
We show that for all integers m >= 4 there exists a 2m x 2m x m latin cuboid that cannot be completed to a 2mx2mx2m latin cube. We also show that for all even m > 2 there exists a (2m-1) x (2m-1) x (m-1) latin cuboid that cannot be extended to any (2m-1) x (2m-1) x m latin cuboid
Solving Problems on Graphs of High Rank-Width
A modulator of a graph G to a specified graph class H is a set of vertices
whose deletion puts G into H. The cardinality of a modulator to various
tractable graph classes has long been used as a structural parameter which can
be exploited to obtain FPT algorithms for a range of hard problems. Here we
investigate what happens when a graph contains a modulator which is large but
"well-structured" (in the sense of having bounded rank-width). Can such
modulators still be exploited to obtain efficient algorithms? And is it even
possible to find such modulators efficiently?
We first show that the parameters derived from such well-structured
modulators are strictly more general than the cardinality of modulators and
rank-width itself. Then, we develop an FPT algorithm for finding such
well-structured modulators to any graph class which can be characterized by a
finite set of forbidden induced subgraphs. We proceed by showing how
well-structured modulators can be used to obtain efficient parameterized
algorithms for Minimum Vertex Cover and Maximum Clique. Finally, we use
well-structured modulators to develop an algorithmic meta-theorem for deciding
problems expressible in Monadic Second Order (MSO) logic, and prove that this
result is tight in the sense that it cannot be generalized to LinEMSO problems.Comment: Accepted at WADS 201
Between proper and strong edge-colorings of subcubic graphs
In a proper edge-coloring the edges of every color form a matching. A
matching is induced if the end-vertices of its edges induce a matching. A
strong edge-coloring is an edge-coloring in which the edges of every color form
an induced matching. We consider intermediate types of edge-colorings, where
edges of some colors are allowed to form matchings, and the remaining form
induced matchings. Our research is motivated by the conjecture proposed in a
recent paper of Gastineau and Togni on S-packing edge-colorings (On S-packing
edge-colorings of cubic graphs, Discrete Appl. Math. 259 (2019), 63-75)
asserting that by allowing three additional induced matchings, one is able to
save one matching color. We prove that every graph with maximum degree 3 can be
decomposed into one matching and at most 8 induced matchings, and two matchings
and at most 5 induced matchings. We also show that if a graph is in class I,
the number of induced matchings can be decreased by one, hence confirming the
above-mentioned conjecture for class I graphs
Orientations, lattice polytopes, and group arrangements II: Modular and integral flow Polynomials of graphs
We study modular and integral flow polynomials of graphs by means of subgroup arrangements and lattice polytopes. We introduce an Eulerian equivalence relation on orientations, flow arrangements, and flow polytopes; and we apply the theory of Ehrhart polynomials to obtain properties of modular and integral flow polynomials. The emphasis is on the geometrical treatment through subgroup arrangements and Ehrhart polynomials. Such viewpoint leads to a reciprocity law on the modular flow polynomial, which gives rise to an interpretation on the values of the modular flow polynomial at negative integers and answers a question by Beck and Zaslavsky.Regal Entertainment Group (Competitive Earmarked Research Grants 600703)Regal Entertainment Group (Competitive Earmarked Research Grants 600506)Regal Entertainment Group (Competitive Earmarked Research Grants 600608
Quasi Polymatroidal Flow Networks
In this paper we give a flow model on directed multigraphs by introducing reflexions of generalized polymatroids at vertices as constraints for the flow conservation. This model has the essential features of the classical flow model, primarily the max-flow min-cut theorem and the polynomial algorithm for computing the maximal feasible (integral) flow
Cubic graphs without a Petersen minor have nowhere-zero 5-flows
We show that every bridgeless cubic graph without a Petersen minor has a nowhere-zero 5-flow. This approximates the known 4-flow conjecture of Tutte
Five Cycle Double Covers of Some Cubic Graphs
AbstractThe main result of this paper can be roughly described as follows. Any bridgeless cubic graph G having a 2-factor with at most two odd components has a 5-cycle double cover, ie., there exists a collection L of five Eulerian subgraphs of G such that every edge of G is an edge of exactly two subgraphs in L. This generalizes and improves several known results. For instance, we can show that any graph with a Hamilton path has a 5-cycle double cover