A sequence d = (d1, d2, · · · , dn) is graphic if there is a simple graph G with degree sequence d, and such a graph G is called a realization of d. A graphic sequence d is linehamiltonian if d has a realizationG such that L(G) is hamiltonian, and is supereulerian if d has a realization G with a spanning eulerian subgraph. In this paper, it is proved that a nonincreasing graphic sequence d = (d...

We consider Cheeger Inequalities for general edge-weighted directed graphs. Previously the directed case was considered by Chung for a probability transition matrix corresponding to a strongly connected graph with weights induced by a stationary distribution. An Eulerian property of these special weights reduces these instances to the undirected case, for which recent results on multi-way spect...

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...

We study a hierarchy of five classes of bijections between the edge sets of two graphs: weak maps, strong maps, cyclic maps, orientable cyclic maps, and chromatic maps. Each of these classes contains the next one and is a natural class of mappings for some family of matroids. For example, f: E(G) --t E(H) is cyclic if every cycle (eulerian subgraph) of G is mapped onto a cycle of H. This class ...

The Brualdi-Shen Conjecture on Eulerian Bipartite Tournaments states that any such graph can be decomposed into oriented 4-cycles. In this article we prove the balanced case of the mentioned conjecture. We show that for any 2n×2n bipartite graph G = (U ∪V,E) in which each vertex has n-neighbors with biadjacency matrixM (or its transpose), there is a particular proper edge coloring of a column p...

We obtain three results concerning the topological cycle space C of a locally finite connected graph G. Confirming a conjecture of Diestel we show that through every edge set E ∈ C there is a topological Euler tour, a continuous map from the circle S to the end compactification |G| of G that traverses every edge in E exactly once and traverses no other edge. Second, we show that for every seque...

We describe and analyze a new construction that produces new Eulerian lattices from old ones. It specializes to a construction that produces new strongly regular cellular spheres (whose face lattices are Eulerian). The construction does not always specialize to convex polytopes; however, in a number of cases where we can realize it, it produces interesting classes of polytopes. Thus we produce ...

The geometric dual of a cellularly embedded graph is fundamental concept in theory and also appears many other branches mathematics. partial an essential generalization which can be obtained by forming the with respect to only subset edges graph. Twisted duality further from combining Petrials duals. Ribbon graphs are new equivalent form old graphs. In this paper, we first characterize regular ...

The field of ethnomathematics is the study of mathematics in the works of art of various cultures [2, 3, 9, 13]. The concepts in this paper are inspired by the visual art of sand drawings that has developed independently in different forms in diverse cultures. Generally speaking, the artist draws a set of dots on some flat surface (usually in the sand or in powder on the floor) and then draws o...

A transition system T of an Eulerian graph G is a family of partitions of the edges incident to each vertex of G into transitions i.e. subsets of size two. A circuit decomposition C of G is compatible with T if no pair of adjacent edges of G is both a transition of T and consecutive in a circuit of C. We give a conjectured characterization of when a 4-regular graph has a transition system which...