On Cyclic Orthogonal Double Covers of Circulant Graphs by Certain Graphs (original) (raw)

On Cartesian Products of Cyclic Orthogonal Double Covers of Circulants

Journal of Mathematics Research, 2014

A collection G of isomorphic copies of a given subgraph G of T is said to be orthogonal double cover (ODC) of a graph T by G, if every edge of T belongs to exactly two members of G and any two different elements from G share at most one edge. An ODC G of T is cyclic (CODC) if the cyclic group of order |V(T)| is a subgroup of the automorphism group of G. In this paper, the CODCs of infinite regular circulant graphs by certain infinite graph classes are considered, where the circulant graphs are labelled by the Cartesian product of two abelian groups.

Orthogonal double covers of complete bipartite graphs by the union of a cycle and a star

2009

Let H be a graph on n vertices and G a collection of n subgraphs of H, one for each vertex. Then G is an orthogonal double cover (ODC) of H if every edge of H occurs in exactly two members of G and any two members of G share exactly an edge whenever the corresponding vertices are adjacent in H. If all subgraphs in G are isomorphic to a given spanning subgraph G, then G is said to be an ODC of H by G. We construct ODCs of H = Kn,n by G = Cm ∪v Sn−m (union of a cycle Cm and a star Sn−m whose center vertex v belongs to that cycle and m = 6, 8, 10, 12 and m < n). Furthermore, we construct ODCs of H = Kn,n by G = Cm∪Sn−m (disjoint union of a cycle and a star) where m = 4, 8 and m < n. In all cases, G is a symmetric starter of the cyclic group of order n. In addition, we introduce a generalization of this result.

On covers of graphs

Discrete Mathematics, 2004

We consider two problems related to the cycle double cover (CDC) conjecture for graphs: the oriented perfect path double cover (OPPDC) and the oriented faithful cycle cover. While we characterize the latter the OPPDC is an interesting open problem.

On the cyclic decomposition of circulant graphs into almost

2011

It is known that if an almost bipartite graph G with n edges possesses a γlabeling, then the complete graph K 2nx+1 admits a cyclic G-decomposition. We introduce a variation of γ-labeling and show that whenever an almost bipartite graph G admits such a labeling, then there exists a cyclic Gdecomposition of a family of circulant graphs. We also determine which odd length cycles admit the variant labeling.

Canonical double covers of circulants

Journal of Combinatorial Theory, Series B, 2022

The canonical double cover B(X) of a graph X is the direct product of X and K 2. If Aut(B(X)) ∼ = Aut(X) × Z 2 then X is called stable; otherwise X is called unstable. An unstable graph is nontrivially unstable if it is connected, non-bipartite and distinct vertices have different neighborhoods. Circulant is a Cayley graph on a cyclic group. Qin et al. conjectured in [J. Combin. Theory Ser. B 136 (2019), 154-169] that there are no nontrivialy unstable circulants of odd order. In this paper we prove this conjecture.

Small Oriented Cycle Double Cover of Graphs

Bulletin of the Malaysian Mathematical Sciences Society, 2015

A small oriented cycle double cover (SOCDC) of a bridgeless graph G on n vertices is a collection of at most n − 1 directed cycles of the symmetric orientation, G s , of G such that each arc of G s lies in exactly one of the cycles. It is conjectured that every 2-connected graph except two complete graphs K 4 and K 6 has an SOCDC. In this paper, we study graphs with SOCDC and obtain some properties of the minimal counterexample to this conjecture.

On the cyclic decomposition of circulant graphs into almost-bipartite graphs

Australas. J Comb., 2011

It is known that if an almost bipartite graph G with n edges possesses a γlabeling, then the complete graphK2nx+1 admits a cyclicG-decomposition. We introduce a variation of γ-labeling and show that whenever an almost bipartite graph G admits such a labeling, then there exists a cyclic Gdecomposition of a family of circulant graphs. We also determine which odd length cycles admit the variant labeling.