Balister [Combin. Probab. Comput. 12 (2003), 1–15] gave a necessary and sufficient condition for a complete multigraph Kn to be arbitrarily decomposable into closed trails of prescribed lengths. In this article we solve the corresponding problem showing that the multigraphs Kn are arbitrarily decomposable into open trails.

Brain graphs (i.e, connectomes) constructed from medical scans such as magnetic resonance imaging (MRI) have become increasingly important tools to characterize the abnormal changes in human brain. Due high acquisition cost and processing time of multimodal MRI, existing deep learning frameworks based on Generative Adversarial Network (GAN) focused predicting missing images a few modalities. Wh...

Not long ago, Bagga, Beineke, and Varma [1] defined the super line multigraph of a simple graph Γ = (V,E) to be the graph Mr(Γ) whose vertex set is Pr(E), the class of r-element subsets of the edge set, and with an adjacency R ∼ R′ (where R,R′ ∈ Pr(E)) for every edge pair (e, f) with e ∈ R and f ∈ R′ such that e and f are adjacent in Γ. Thus, the number of edges joining R and R′ in Mr(Γ) is the...

The paper studies edge-coloring of signed multigraphs and extends classical Theorems Shannon K\"onig to multigraphs. We prove that the chromatic index a multigraph $(G,\sigma_G)$ is at most $\lfloor \frac{3}{2} \Delta(G) \rfloor$. Furthermore, balanced $(H,\sigma_H)$ $\Delta(H) + 1$ with $\Delta(H)$ are characterized.

A cycle packing in a (directed) multigraph is a vertex disjoint collection of (directed) elementary cycles. If D is a demiregular multidigraph we show that the arcs of D can be partitioned into Ai. cycle packings where Ain is the maximum indegree of a vertex in D. We then show that the maximum length cycle packings in any digraph contain a common vertex.

In this paper, we give the first polynomial-time algorithm for determining the edge expansion for a graph of fixed orientable genus. We show that for a multigraph G with m edges and orientable genus g, the edge expansion of G can be determined in time m. We show that the same is true for various other similar measures of edge connectivity.

We introduce the concept of list total colorings and prove that every multigraph of maximum degree 3 is 5-total-choosable. We also show that the total-choosability of a graph of maximum degree 2 is equal to its total chromatic number.

A multicircuit is a multigraph whose underlying simple graph is a circuit (a connected 2-regular graph). In this pair of papers, it is proved that every multicircuit C has total choosability (i.e., list total chromatic number) ch00(C ) equal to its ordinary total chromatic number 00(C ). In the present paper, the kernel method is used to prove this for every multicircuit that