Abstract
A dicycle cover of a digraph D is a family F of dicycles of D such that each arc of D lies in at least one dicycle in F. We investigate the problem of determining the upper bounds for the minimum number of dicycles which cover all arcs in a strong digraph. Best possible upper bounds of dicycle covers are obtained in a number of classes of digraphs including strong tournaments, Hamiltonian oriented graphs, Hamiltonian oriented complete bipartite graphs, and families of possibly non-Hamiltonian digraphs obtained from these digraphs via a sequence of 2-sum operations.
Highlights
We consider finite loopless graphs and digraphs, and undefined notations and terms will follow [1] for graphs and [2] for digraphs
Luo and Chen [4] proved that this conjecture holds for 2-connected simple cubic graphs
It has been shown that, for plane triangulations, serial-parallel graphs, or planar graphs in general, one can have a better bound for the number of cycles used in a cover [5,6,7,8]
Summary
We consider finite loopless graphs and digraphs, and undefined notations and terms will follow [1] for graphs and [2] for digraphs. Bondy [3] conjectured that if G is a 2-connected simple graph with n ≥ 3 vertices, G has a cycle cover C with |C| ≤ (2n − 3)/3. Barnette [9] proved that if G is a 3-connected simple planar graph of order n, the edges of G can be covered by at most (n+1)/2 cycles. We will first show that every Hamiltonian oriented graph with n vertices and m arcs can be covered by at most m − n + 1 dicycles. We show that, for every Hamiltonian graph G with n vertices and m edges, there exists an orientation D = D(G) of G such that any dicycle cover of D must have at least m − n + 1 dicycles
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
