A graph G(p, q) is said to be odd harmonious if there exists an injection f : V (G) → {0, 1, 2, · · · , 2q − 1} such that the induced function f∗ : E(G) → {1, 3, · · · , 2q − 1} defined by f∗(uv) = f(u) + f(v) is a bijection. In this paper we prove that the plus graph Pln , open star of plus graph S(t.P ln), path union of plus graph Pln, joining of Cm and plus graph Pln with a path, one point u...

A 2-factor in a graph is a spanning 2-regular subgraph, or equivalently a spanning collection of disjoint cycles. In this paper we investigate the existence of 2-factors with a bounded number of odd cycles in a graph. We extend results of Ryjáček, Saito, and Schelp (Closure, 2-factors, and cycle coverings in claw-free graphs, J. Graph Theory, 32 (1999), no. 2, 109-117) and show that the number ...

Edge (5, 3)-bicolorings of a (5, 3)-regular bipartite graph relating 5cycles and vertices of the Petersen graph lead to solutions of the Great Circle Challenge puzzle. This takes for odd n = 2k + 1 > 1 to a oneto-one correspondence from the family Cn of n-cycles of Kn onto the family Ok of n-cycles of the odd graph Ok+1. It also takes for odd l with 3 ≤ l ≤ ⌊n/2⌋ to a graph having Ok as vertex ...

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...