The concept “odd-minor” which is a generalization of minor-relation has received considerable amount of attention by many researchers, and led to several beautiful conjectures and results. We say that H has an odd complete minor of order l if there are l vertex disjoint trees in H such that every two of them are joined by an edge, and in addition, all the vertices of trees are two-colored in su...

For terms not defined here, the reader is referred to [6]. For a graph G=(V, E), we denote by G the complement of G. We use |(G) to denote the size of a largest clique in G and :(G) to denote the size of a largest stable set in G, or simply | and : when no confusion is possible. A graph G is said to be perfect if, for each induced subgraph H of G, H can be coloured with |(H) colours such that e...

It is proved that if G is a k-connected graph which does not contain K;; with k being odd, then G has an edge e such that the graph obtained from G by contracting e is still k-connected. The same conclusion does not hold when k is even. This result is a generalization of the famous theorem of Thomassen [J. Graph Theory 5 (1981), 351--354] when k is odd.

We show that almost every Cayley graph Γ of an abelian group G of odd prime-power order has automorphism group as small as possible. Additionally, we show that almost every Cayley (di)graph Γ of an abelian group G of odd prime-power order that does not have automorphism group as small as possible is a normal Cayley (di)graph of G (that is, GL/Aut(Γ)).

A proof is given of the result about binary matroids that implies that a connected graph is Eulerian if and only if every edge lies in an odd number of circuits, and a graph is bipartite if and only if every edge lies in an odd number of cocircuits (minimal cutsets). A proof is also given of the result that the edge set of every graph can be expressed as a disjoint union of circuits and cocircu...

We give a characterization of a current assignment on the bipartite Möbius ladder graph with 2n + 1 rungs. Such an assignment yields an index one current graph with current group Z12n+7 that generates an orientable face 2-colorable triangular embedding of the complete graph K12n+7 or, equivalently, an orientable biembedding of two cyclic Steiner triple systems of order 12n+7. We use our charact...

A normal odd partition T of the edges of a cubic graph is a partition into trails of odd length (no repeated edge) such that each vertex is the end vertex of exactly one trail of the partition and internal in some trail. For each vertex v, we can distinguish the edge for which this vertex is pending. Three normal odd partitions are compatible whenever these distinguished edges are distinct for ...