Degree Sequences of Edge-Colored Graphs in Specified Families and Related Problems

Movement has been made in recent times to generalize the study of degree sequences to k-edge-colored graphs and doing so requires the notion of a degree vector\u3c\italic\u3e. The degree vector of a vertex v\u3c\italic\u3e in a k-edge-colored graph is a column vector in which entry i\u3c\italic\u3e indicates the number of edges of color i\u3c\italic\u3e incident to v\u3c\italic\u3e. Consider the following question which we refer to as the \u3c\italic\u3ek-Edge-Coloring Problem\u3c\italic\u3e. Given a set of column vectors C\u3c\italic\u3e and a graph family F\u3c\italic\u3e, when does there exist some k-edge-colored graph in F\u3c\italic\u3e whose set of degree vectors is C\u3c\italic\u3e? This question is NP-Complete in general but certain graph families yield tractable results. In this document, I present results on the k-Edge-Coloring Problem and the related Factor Problem for the following families of interest: unicyclic graphs, disjoint unions of paths (DUPs), disjoint union of cycles (DUCs), grids, and 2-trees