Circuit covers of cubic signed graphs

Characteristic Flows on Signed Graphs and Short Circuit Covers

We generalise to signed graphs a classical result of Tutte [Canad. J. Math. 8 (1956), 13–28] stating that every integer flow can be expressed as a sum of characteristic flows of circuits. In our generalisation, the rôle of circuits is taken over by signed circuits of a signed graph which are known to occur in two types – either balanced circuits or pairs of disjoint unbalanced circuits connecte...

Signed B-edge Covers of Graphs

We study a signed variant of edge covers of graphs. Let b be a positive integer, and let G be a graph with minimum degree at least b. A signed b-edge cover of G is a function f : E(G) → {−1, 1} satisfying e∈EG(v) f(e) ≥ b for every v ∈ V (G). The minimum of the values of e∈E(G) f(e), taken over all signed b-edge covers f of G, is called the signed b-edge cover number and is denoted by ρb(G). Fo...

Signed (b, k)-Edge Covers in Graphs

Let be a simple graph with vertex set and edge set . Let have at least vertices of degree at least , where and b are positive integers. A function is said to be a signed -edge cover of G if G ( ) V G ( ) e E v ( ) E G G : ( f E k b k ) { 1,1} G   ( , ) b k ( ) f e b    for at least vertices of , where . The value k v G ( ) = {uv E( ( ) E v G u N v   ) | } ( ) min ( ) G e E f e   , taki...

6-cycle Double Covers of Cubic Graphs

A cycle double cover (CDC) of an undirected graph is a collection of the graph’s cycles such that every edge of the graph belongs to exactly two cycles. We describe a constructive method for generating all the cubic graphs that have a 6-CDC (a CDC in which every cycle has length 6). As an application of the method, we prove that all such graphs have a Hamiltonian cycle. A sense of direction is ...

Circuit Covers in Series-Parallel Mixed Graphs