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 \(R(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{d} ,L(m))\).
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References
V. Chungphaisan, Conditions for sequences to be r-graphic, Discrete Math. 7 (1974), 31–39.
D.R. Fulkerson, A.J. Hoffman, and M.H. McAndrew, Some properties of graphs with multiple edges, Canad. J. Math. 17 (1965), 166–177.
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© 1981 Springer-Verlag
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Billington, D. (1981). Connected subgraphs of the graph of multigraphic realisations of a degree sequence. In: McAvaney, K.L. (eds) Combinatorial Mathematics VIII. Lecture Notes in Mathematics, vol 884. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091814
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DOI: https://doi.org/10.1007/BFb0091814
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