Using results by McKee and Woodall on binary matroids, we prove that the set of postman sets has odd cardinality, generalizing a result by Toida on the cardinality of cycles in Eulerian graphs. We study the relationship between T-joins and blocks of the underlying graph, obtaining a decomposition of postman sets in terms of blocks. We conclude by giving several characterizations of T-joins whic...

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 30 March 2021Accepted: 24 September 2021Published online: 06 January 2022Keywordshypergraphs, Eulerian and Hamiltonian graphs, Absorption MethodAMS Subject Headings05C65, 05C45Publication DataISSN (print): 0895-4801ISSN (online): 1095-7146Publisher: Society for Industrial...

We give new sufficient conditions for a sequence of polynomials to have only real zeros based on the method of interlacing zeros. As applications we derive several well-known facts, including the reality of zeros of orthogonal polynomials, matching polynomials, Narayana polynomials and Eulerian polynomials. We also settle certain conjectures of Stahl on genus polynomials by proving them for cer...

We prove that every Eulerian orientation of Km,n contains 1 4+ √ 8 mn(1 − o(1)) arc-disjoint directed 4-cycles, improving earlier lower bounds. Combined with a probabilistic argument, this result is used to prove that every regular tournament with n vertices contains 1 8+ √ 32 n2(1−o(1)) arc-disjoint directed 4-cycles. The result is also used to provide an upper bound for the distance between t...

In this note we construct a family of SAT-instance based on Eulerian graphs which are aimed at being hard for resolution based SAT-solvers. We discuss some experiments made with instances of this type and how a solver can try to avoid at least some of the pitfalls presented by these instances. Finally we look at how the density of subformulae can influence the hardness of SAT instances.

Let G be an Eulerian bipartite digraph with vertex partition sizes m,n. We prove the following Turán-type result: If e(G) > 2mn/3 then G contains a directed cycle of length at most 4. The result is sharp. We also show that if e(G) = 2mn/3 and no directed cycle of length at most 4 exists, then G must be biregular. We apply this result in order to obtain an improved upper bound for the diameter o...

In this paper, we consider the asymptotic properties of the numbers of spanning trees and Eulerian trials in circulant digraphs and graphs. Let C(p, s1, s2, . . . , sk) be a directed or undirected circulant graph. Let T (C(p, s1, s2, . . . , sk)) and E(C(p, s1, s2, . . . , sk)) be the numbers ∗This work was partially supported by the Natural Science Foundation of China, email: fjzhang@jingxian....

A graph G is supereulerian if it has a spanning eulerian subgraph. We prove that every 3-edge-connected graph with the circumference at most 11 has a spanning eulerian subgraph if and only if it is not contractible to the Petersen graph. As applications, we determine collections F1, F2 and F3 of graphs to prove each of the following (i) Every 3-connected {K1,3, Z9}-free graph is hamiltonian if ...

A conformally indecomposable flow f on a signed graph Σ is a nonzero integral flow that cannot be decomposed into f = f1 + f2, where f1, f2 are nonzero integral flows having the same sign (both ≥ 0 or both ≤ 0) at every edge. This paper is to classify at integer scale conformally indecomposable flows into characteristic vectors of Eulerian cycle-trees — a class of signed graphs having a kind of...

The windy postman problem is the NP-hard problem of finding the minimum cost of a tour traversing all edges of an undirected graph, where the cost of traversal of an edge depends on the direction. Given an undirected graph G, we consider the polyhedron O(G) induced by the linear programming relaxation of a well-known integer programming formulation of the problem. We say that G is windy postman...