Decompositions of graphs into cycles with chords

Decompositions of Graphs into 5-Cycles and Other Small Graphs

In this paper we consider the problem of finding the smallest number q such that any graph G of order n admits a decomposition into edge disjoint copies of a fixed graph H and single edges with at most q elements. We solve the case when H is the 5-cycle, the 5-cycle with a chord and any connected non-bipartite non-complete graph of order 4.

Chords of Longest Cycles in Cubic Graphs

We describe a general sufficient condition for a Hamiltonian graph to contain another Hamiltonian cycle. We apply it to prove that every longest cycle in a 3-connected cubic graph has a chord. We also verify special cases of an old conjecture of Sheehan on Hamiltonian cycles in 4-regular graphs and a recent conjecture on a second Hamiltonian cycle by Triesch, Nolles, and Vygen. 1997 Academic Press

Cycle decompositions V: Complete graphs into cycles of arbitrary lengths

We show that the complete graph on n vertices can be decomposed into t cycles of specified lengths m1, . . . ,mt if and only if n is odd, 3 ≤ mi ≤ n for i = 1, . . . , t, and m1+· · ·+mt = ( n 2 ) . We also show that the complete graph on n vertices can be decomposed into a perfect matching and t cycles of specified lengths m1, . . . ,mt if and only if n is even, 3 ≤ mi ≤ n for i = 1, . . . , t...

Further decompositions of complete tripartite graphs into 5-cycles

Decompositions of complete tripartite graphs into k-cycles

We show that a complete tripartite graph with three partite sets of equal size m may be decomposed into k-cycles for any k 2: 3 if and only if k divides 3m2 and k ::; 3m.