Let be a family of -uniform hypergraphs (henceforth -graphs). An -saturated -graph is a maximal -graph not containing any member of as a subgraph. For integers , let be the collection of all
Let be a family of -uniform hypergraphs (henceforth -graphs). An -saturated -graph is a maximal -graph not containing any member of as a subgraph. For integers , let be the collection of all
Given a hypergraph , the dual hypergraph of is the hypergraph of all minimal transversals of . The dual hypergraph is always Sperner, that is, no hyperedge contains another. A special case of Sperner hypergraphs are the conformal Sperner hypergraphs, which correspond to the families of maximal cliques of graphs. All these notions play an important role in many fields of mathematics and computer science, including combinatorics, algebra, database theory, and so on. Motivated by a question related to clique transversals in graphs, we study in this paper conformality of dual hypergraphs and prove several results related to the problem of recognizing this property. In particular, we show that the problem is in co-NP and that it can be solved in polynomial time for hypergraphs of bounded dimension. For dimension 3, we show that the problem can be reduced to 2-Satisfiability. Our approach has an application in algorithmic graph theory: we obtain a polynomial-time algorithm for recognizing graphs in which all minimal transversals of maximal cliques have size at most , for any fixed .
The total face color polynomial is based upon the Poincaré polynomials of a family of filtered -color homologies. It counts the number of -face colorings of ribbon graphs for each positive integer . As such, it may be seen as a successor of the Penrose polynomial, which at counts 3-edge colorings (and consequently 4-face colorings) of planar trivalent graphs. In this paper, we describe a state sum formula for the polynomial. This formula unites two different perspectives about graph coloring: one based upon topological quantum field theory and the other on diagrammatic tensors.
In infinite graph theory, the notion of ends, first introduced by Freudenthal and Jung for locally finite graphs, plays an important role when generalizing statements from finite graphs to infinite ones. Nash-Williams' Tree-Packing Theorem and MacLane's Planarity Criteria are examples of results that allow a topological approach, in which ends may be considered as endpoints of rays. In fact, there are extensive studies in the literature showing that classical (vertex-)connectivity theorems for finite graphs can be discussed regarding ends, in a more general context. However, aiming to generalize results of edge-connectivity, this paper recalls the definition of edge-ends in infinite graphs due to Hahn, Laviolette and Širáň. In terms of that object, we state an edge version of Menger's Theorem (following a previous work of Polat) and generalize the Lovász-Cherkassky Theorem for infinite graphs with edge-ends (inspired by a recent paper of Jacobs, Joó, Knappe, Kurkofka and Melcher).
We show that the basis graph of an even delta-matroid is Hamiltonian if it has more than two vertices. More strongly, we prove that for two distinct edges and sharing a common end, it has a Hamiltonian cycle using and avoiding unless it has at most two vertices or it is a cycle of length at most four. We also prove that if the basis graph is not a hypercube graph, then each vertex belongs to cycles of every length , and each edge belongs to cycles of every length . For the last theorem, we provide two proofs, one of which uses the result of Naddef (1984) on polytopes and the result of Chepoi (2007) on basis graphs of even delta-matroids, and the other is a direct proof using various properties of even delta-matroids. Our theorems generalize the analogous results for matroids by Holzmann and Harary (1972) and Bondy and Ingleton (1976).
A long-standing conjecture of Berge suggests that every bridgeless cubic graph can be expressed as a union of at most five perfect matchings. This conjecture trivially holds for 3-edge-colourable cubic graphs, but remains widely open for graphs that are not 3-edge-colourable. The aim of this paper is to verify the validity of Berge's conjecture for cubic graphs that are in a certain sense close to 3-edge-colourable graphs. We measure the closeness by looking at the colouring defect, which is defined as the minimum number of edges left uncovered by any collection of three perfect matchings. While 3-edge-colourable graphs have defect 0, every bridgeless cubic graph with no 3-edge-colouring has defect at least 3. In 2015, Steffen proved that the Berge conjecture holds for cyclically 4-edge-connected cubic graphs with colouring defect 3 or 4. Our aim is to improve Steffen's result in two ways. We show that all bridgeless cubic graphs with defect 3 satisfy Berge's conjecture irrespectively of their cyclic connectivity. If, additionally, the graph in question is cyclically 4-edge-connected, then four perfect matchings suffice, unless the graph is the Petersen graph. The result is best possible as there exists an infinite family of cubic graphs with cyclic connectivity 3 which have defect 3 but cannot be covered with four perfect matchings.
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：