The concept of super line graph was introduced in the year 1995 by Bagga, Beineke and Varma. Given a graph with at least r edges, the super line graph of index r, Lr(G), has as its vertices the sets of r-edges of G, with two adjacent if there is an edge in one set adjacent to an edge in the other set. The line completion number lc(G) of a graph G is the least positive integer r for which Lr(G) ...

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...

We introduce a family of multi-way Cheeger-type constants {h k , k = 1, 2, . . . , N} on a signed graph Γ = (G, σ) such that h k = 0 if and only if Γ has k balanced connected components. These constants are switching invariant and bring together in a unified viewpoint a number of important graph-theoretical concepts, including the classical Cheeger constant, the non-bipartiteness parameter of D...

Graph coverings are known to induce surjections of their critical groups. Here we describe the kernels of these morphisms in terms of data parametrizing the covering. Regular coverings are parametrized by voltage graphs, and the above kernel can be identified with a naturally defined voltage graph critical group. For double covers, the voltage graph is a signed graph, and the theory takes a par...

A signed bipartite graph G(U, V ) is a bipartite graph in which each edge is assigned a positive or a negative sign. The signed degree of a vertex x in G(U, V ) is the number of positive edges incident with x less the number of negative edges incident with x. The set S of distinct signed degrees of the vertices of G(U, V ) is called its signed degree set. In this paper, we prove that every set ...

A signed bipartite graph G(U, V) is a bipartite graph in which each edge is assigned a positive or a negative sign. The signed degree of a vertex x in G(U, V) is the number of positive edges incident with x less the number of negative edges incident with x. The set S of distinct signed degrees of the vertices of G(U, V) is called its signed degree set. In this paper, we prove that every set of ...