Connected subgraphs of the graph of multigraphic realisations of a degree sequence

Abstract

An m-graph is a graph, without loops, but with multiple edges of any multiplicity less than or equal to m. An exact m-graph is an m-graph with at least one edge of multiplicity m. A new proof is given that the graph $R(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{d} ,L(m))$, of all m-graphic realisations of a degree sequence, $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{d}$, is connected. This is done by taking any two vertices of $R(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{d} ,L(m))$, say G and H, and finding a path between them which preserves any previously chosen edge of multiplicity m that occurs in both G and H. The construction of this path also establishes best possible upper and lower bounds on the length of the shortest path between any two vertices of $R(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{d} ,L(m))$.