Graph whose edges are in small cycles

Hamiltonian cycles avoiding sets of edges in a graph

Let G be a graph and H be a subgraph of G. If G contains a hamiltonian cycle C such that E(C)∩E(H) is empty, we say that C is an H-avoiding hamiltonian cycle. Let F be any graph. If G contains an H-avoiding hamiltonian cycle for every subgraph H of G such that H ∼= F , then we say that G is F -avoiding hamiltonian. In this paper, we give minimum degree and degree-sum conditions which ensure tha...

Vertex-disjoint cycles containing specified edges in a bipartite graph

Dirac and Ore-type degree conditions are given for a bipartite graph to contain vertex disjoint cycles each of which contains a previously specified edge. This solves a conjecture of Wang in [6].

Covering a bipartite graph with cycles passing through given edges

We propose a conjecture: for each integer k 2:: 2, there exists N (k) such that if G = (Vb \12; E) is a bipartite graph with IV11 = 1\121 = n 2: N(k) and d( x) + d(y) 2: n + k for each pair of non-adjacent vertices x and y of G with x E V1 and y E \12, then for any k independent edges el, ... , ek of G, there exist k vertex-disjoint cycles G1, ... , Gk in G such that ei E E(Gi ) for all i E {I,...

On Hamiltonian Cycles through Prescribed Edges of a Planar Graph

We use [3] for terminology and notation not defined here and consider finite simple graphs only. The first major result on the existence of hamiltonian cycles in graphs embeddable in surfaces was by H. Whitney [12] in 1931, who proved that 4-connected maximal planar graphs are hamiltonian. In 1956, W.T. Tutte [10,11] generalized Whitney’s result from maximal planar graphs to arbitrary 4-connect...

The Maximum Number of Edges in a Strongly Multiplicative Graph