Cycle systems in the complete bipartite graph minus a one-factor

AbstractLet Kn,n−I denote the complete bipartite graph with n vertices in each part from which a 1-factor I has been removed. An m-cycle system of Kn,n−I is a collection of m-cycles whose edges partition Kn,n−I. Necessary conditions for the existence of such an m-cycle system are that m⩾4 is even, n⩾3 is odd, m⩽2n, and m|n(n−1). In this paper, we show these necessary conditions are sufficient except possibly in the case that m≡0(mod4) with n<m<2n

A Kuratowski theorem for nonorientable surfaces

AbstractLet Σ denote a surface. A graph G is irreducible for Σ provided that G does not embed in Σ, but any proper subgraph does so embed. Let I(Σ) denote the set of graphs without degree two vertices which are irreducible for Σ. Observe that a graph embeds in Σ if and only if it does not contain a subgraph homeomorphic to a member of I(Σ). For example, Kuratowski's theorem shows that I(Σ) = {K3,3, K5} when Σ is the sphere. In this paper we prove that the set I(Σ) is finite for each nonorientable surface, setting in part a conjecture of Erdös from the 1930s

Trinity symmetry and kaleidoscopic regular maps

A cellular embedding of a connected graph (also known as a map) on an orientable surface has trinity symmetry if it is isomorphic to both its dual and its Petrie dual. A map is regular if for any two incident vertex-edge pairs there is an automorphism of the map sending the first pair onto the second. Given a map M with all vertices of the same degree d, for any e relatively prime to d the power map Me is formed from M by replacing the cyclic rotation of edges at each vertex on the surface with the e th power of the rotation. A map is kaleidoscopic if all of its power maps are pairwise isomorphic. In this paper, we present a covering construction that gives infinite families of kaleidoscopic regular maps with trinity symmetry

Halin's theorem for cubic graphs on an annulus

Halin's Theorem characterizes those locally-finite, infinite graphs that embed in the plane without accumulation points by giving a set of six topologically excluded subgraphs. We prove the analogous theorem for cubic graphs that embed in an annulus without accumulation points, finding the complete set of 29 excluded subgraphs

Representing graphs in Steiner triple systems

Let G = (V, E) be a simple graph and let T = (P, B) be a Steiner triple system. Let φ be a one-to-one function from V to P. Any edge e = {u, v} has its image {φ(u), φ(v)} in a unique block in B. We also denote this induced function from edges to blocks by φ. We say that T represents G if there exists a one-to-one function φ : V → P such that the induced function φ : E → B is also one-to-one; that is, if we can represent vertices of the graph by points of the triple system such that no two edges are represented by the same block. In this paper we examine when a graph can be represented by an STS. First, we find a bound which ensures that every graph of order n is represented in some STS of order f(n). Second, we find a bound which ensures that every graph of order n is represented in every STS of order g(n). Both of these answers are related to finding an independent set in an STS. Our question is a generalization of finding such independent sets. We next examine which graphs can be represented in STS’s of small orders. Finally, we give bounds on the orders of STS’s that are guaranteed to embed all graphs of a given maximum degree

Two Graphs Without Planar Covers

In this note we prove that two speci c graphs do not have nite planar covers. The graphs are K 7 C 4 and K 4;5 4K 2 . This research is related to Negami&apos;s 1-2-1 Conjecture which states \A graph G has a nite planar cover if and only if it embeds in the projective plane&quot;. In particular