This paper is motivated by a survey on the existence of G-designs by Adams, Bryant and Buchanan, where they gave the spectrum of the decomposition of complete graphs into graphs with small numbers of vertices. We give difference family-type constructions to decompose λ copies, where λ ≥ 2, of the complete graph K v into several multigraphs on four vertices and five edges including so called " 2...

For n ≥ 4, the complete n-vertex multidigraph with arc multiplicity λ is proved to have a decomposition into directed paths of arbitrarily prescribed lengths ≤ n−1 and different from n−2, unless n = 5, λ = 1, and all lengths are to be n−1 = 4. For λ = 1, a more general decomposition exists; namely, up to five paths of length n− 2 can also be prescribed.

A graph is k-linked (k-edge-linked), k ≥ 1, if for each k pairs of vertices x1, y1, · · · , xk, yk, there exist k pairwise vertex-disjoint (respectively edge-disjoint) paths, one per pair xi and yi, i = 1, 2, · · · , k. Here we deal with the properly-edge-colored version of the k-linked (kedge-linked) problem in edge-colored graphs. In particular, we give conditions on colored degrees and/or nu...

A graph G is 1-Hamilton-connected if G − x is Hamilton-connected for every vertex x ∈ V (G). In the paper we introduce a closure concept for 1-Hamiltonconnectedness in claw-free graphs. If G is a (new) closure of a claw-free graph G, then G is 1-Hamilton-connected if and only if G is 1-Hamilton-connected, G is the line graph of a multigraph, and for some x ∈ V (G), G − x is the line graph of a ...

Due to some intractability considerations, reasonable formulation of necessary and sufficient conditions for decomposability of a general multigraph G into a fixed connected multigraph H , is probably not feasible if the underlying simple graph of H has three or more edges. We study the case where H consists of two underlying edges. We present necessary and sufficient conditions for H-decomposa...

For a finite multigraph G, the reliability function of G is the probability RG(q) that if each edge of G is deleted independently with probability q then the remaining edges of G induce a connected spanning subgraph of G; this is a polynomial function of q. In 1992, Brown and Colbourn conjectured that for any connected multigraph G, if q ∈ C is such that RG(q) = 0 then |q| ≤ 1. We verify that t...

For a fixed multigraph H with vertices w1, . . . , wm, a graph G is H-linked if for every choice of vertices v1, . . . , vm in G, there exists a subdivision of H in G such that vi is the branch vertex representing wi (for all i). This generalizes the notions of k-linked, k-connected, and k-ordered graphs. Given a connected multigraph H with k edges and minimum degree at least two and n 7.5k, we...

A subgraph H of a multigraph G is called strongly spanning, if any vertex of G is not isolated in H. H is called maximum k-edge-colorable, if H is proper k-edge-colorable and has the largest size. We introduce a graph-parameter sp(G), that coincides with the smallest k for which a multigraph G has a maximum k-edge-colorable subgraph that is strongly spanning. Our first result offers some altern...

The connectional brain template (CBT) is a compact representation (i.e., single connectivity matrix) multi-view networks of given population. CBTs are especially very powerful tools in dysconnectivity diagnosis as well holistic mapping if they learned properly – i.e., occupy the center Even though accessing large-scale datasets much easier nowadays, it still challenging to upload all these clin...