In 1992 Gyárfás showed that a graph G having only k odd cycle lengths is (2k + 1)colourable, if it does not contain a K2k+2. In this paper, we will present the results for graphs containing only odd cycles of length 2m − 1 and 2m + 1 as done in [S. Matos Camacho, Colourings of graph with prescribed cycle lengths, diploma thesis, TU Bergakademie Freiberg, 2006. [3]]. We will show that these grap...

We prove that there exists a cyclic Hamiltonian k-cycle system of the complete graph if and only if k is odd but k = 15 and p with p prime and ¿ 1. As a consequence we have the existence of a cyclic k-cycle system of the complete graph on km vertices for any pair (k; m) of odd integers with k as above but (k; m) = (3; 3). c © 2003 Elsevier B.V. All rights reserved.

Characterizing regular covers of edge-transitive or arc-transitive graphs is currently a hot topic in algebraic graph theory. In this paper, we will classify arctransitive regular cyclic covers of the complete bipartite graph Kp,p for each odd prime p. The classification consists of four infinite families of graphs. In particular, such covers exist for each odd prime p. The regular elementary a...

Petru\v{s}evski and \v{S}krekovski \cite{odd9} recently introduced the notion of an odd colouring a graph: proper vertex graph $G$ is said to be \emph{odd} if for each non-isolated $x \in V(G)$ there exists colour $c$ appearing number times in $N(x)$. proved that any planar using at most $9$ colours and, together with Caro \cite{oddremarks}, showed $8$ are enough significant family graphs. We s...

In this paper we examine the problem of decomposing the lexicographic product of a cycle with an empty graph into cycles of uniform length. We determine necessary and sufficient conditions for a solution to this problem when the cycles are of odd length. We apply this result to find necessary and sufficient conditions to decompose a complete equipartite graph into cycles of uniform length, in t...

Let G be a finite group and let $GK(G)$ be the prime graph of G. We assume that $n$ is an odd number. In this paper, we show that if $GK(G)=GK(B_n(p))$, where $ngeq 9$ and $pin {3,5,7}$, then G has a unique nonabelian composition factor isomorphic to $B_n(p)$ or $C_n(p)$ . As consequences of our result, $B_n(p)$ is quasirecognizable by its spectrum and also by a new proof, the ...

In this paper some operations are described that transform every graph into a perfect graph by replacing nodes with sets of new nodes. The transformation is done in such a way that every stable set in the perfect graph corresponds to a stable set in the original graph. These operations yield a purely combinatorial augmentation procedure for finding a maximum weighted stable set in a graph. Star...

Let G be an r-regular graph of order n and independence number α(G). We show that if G has odd girth 2k + 3 then α(G) ≥ n1−1/kr1/k . We also prove similar results for graphs which are not regular. Using these results we improve on the lower bound of Monien and Speckenmeyer, for the independence number of a graph of order n and odd girth 2k + 3. AMS Subject Classification. 05C15 §

Motivated by the work of Nešetřil and Rödl on “Partitions of vertices”, we are interested in obtaining some quantitative extensions of their result. In particular, given a natural number r and a graph G of order m with odd girth g, we show the existence of a graph H with odd girth at least g and order that is polynomial in m such that every r-coloring of the vertices of H yields a monochromatic...

A (p, q) connected graph is edge-odd graceful graph if there exists an injective map f: E(G) → {1, 3, ..., 2q-1} so that induced map f+: V(G) → {0, 1,2, 3, ..., (2k-1)}defined by f+(x) o f(x, y) (mod 2k), where the vertex x is incident with other vertex y and k = max {p, q} makes all the edges distinct and odd. In this article, the Edge-odd gracefulness of C3  Pn and C3  2Pn is obtained. Refe...