4-Cycle Decompositions of Graphs

Even cycle decompositions of 4-regular graphs and line graphs

An even cycle decomposition of a graph is a partition of its edge into even cycles. We first give some results on the existence of even cycle decomposition in general 4-regular graphs, showing that K5 is not the only graph in this class without such a decomposition. Motivated by connections to the cycle double cover conjecture we go on to consider even cycle decompositions of line graphs of 2-c...

On path-cycle decompositions of triangle-free graphs

In this work, we study conditions for the existence of length-constrained path-cycle decompositions, that is, partitions of the edge set of a graph into paths and cycles of a given minimum length. Our main contribution is the characterization of the class of all triangle-free graphs with odd distance at least 3 that admit a path-cycle decomposition with elements of length at least 4. As a conse...

On even cycle decompositions of 4-regular line graphs

We prove that the Petersen colouring conjecture implies a conjecture of Markström saying that the line graph of every bridgeless cubic graph is decomposable into cycles of even length. In addition, we describe two infinite families of 4regular graphs: the first family consists of 3-connected graphs with no even cycle decomposition and the second one consists of 4-connected signed graphs with no...

Cycle decompositions of λ-fold complete equipartite graphs

In this paper we give constructions for various new fixed length cycle decompositions of λ-fold complete equipartite graphs. In particular, we determine necessary and sufficient conditions for the existence of such a decomposition in the case where the cycle length is prime.

On 4-connected graphs without even cycle decompositions

Decompositions of complete multipartite graphs via generalized graceful labelings

We prove the existence of infinite classes of cyclic Γ-decompositions of the complete multipartite graph, Γ being a caterpillar, a hairy cycle or a cycle. All results are obtained by the construction of d-divisible α-labelings of Γ, introduced in [A. Pasotti, On d-graceful labelings, Ars Combin. 111 (2013), 207–223] as a generalization of classical α-labelings. It is known that such labelings i...