Shannon multigraph

In the mathematical discipline of graph theory, Shannon multigraphs, named after Claude Shannon by Vizing (1965), are a special type of triangle graphs, which are used in the field of edge coloring in particular.

A Shannon multigraph is multigraph with 3 vertices for which either of the following conditions holds:

a) all 3 vertices are connected by the same number of edges.
b) as in a) and one additional edge is added.

More precisely one speaks of Shannon multigraph $Sh(n)$ , if the three vertices are connected by $\left\lfloor \frac{n}{2} \right\rfloor$ , $\left\lfloor \frac{n}{2} \right\rfloor$ and $\left\lfloor \frac{n+1}{2} \right\rfloor$ edges respectively. This multigraph has maximum degree $n$ . Its multiplicity (the maximum number of edges in a set of edges that all have the same endpoints) is $\left\lfloor \frac{n+1}{2} \right\rfloor$ .

Examples

Shannon multigraphs
Sh(2)
Sh(3)
Sh(4)
Sh(5)
Sh(6)
Sh(7)

Edge coloring

This nine-edge Shannon multigraph requires nine colors in any edge coloring; its vertex degree is six and its multiplicity is three.

According to a theorem of Shannon (1949), every multigraph with maximum degree $\Delta$ has an edge coloring that uses at most $\frac32\Delta$ colors. When $\Delta$ is even, the example of the Shannon multigraph with multiplicity $\Delta/2$ shows that this bound is tight: the vertex degree is exactly $\Delta$ , but each of the $\frac32\Delta$ edges is adjacent to every other edge, so it requires $\frac32\Delta$ colors in any proper edge coloring.

A version of Vizing's theorem (Vizing 1964) states that every multigraph with maximum degree $\Delta$ and multiplicity $\mu$ may be colored using at most $\Delta+\mu$ colors. Again, this bound is tight for the Shannon multigraphs.

References

Fiorini, S.; Wilson, Robin James (1977), Edge-colourings of graphs, Research Notes in Mathematics 16, London: Pitman, p. 34, ISBN 0-273-01129-4, MR 0543798
Shannon, Claude E. (1949), "A theorem on coloring the lines of a network", J. Math. Physics 28: 148–151, MR 0030203 .
Volkmann, Lutz (1996), Fundamente der Graphentheorie (in German), Wien: Springer, p. 289, ISBN 3-211-82774-9 .
Vizing, V. G. (1964), "On an estimate of the chromatic class of a p-graph", Diskret. Analiz. 3: 25–30, MR 0180505 .
Vizing, V. G. (1965), "The chromatic class of a multigraph", Kibernetika 1965 (3): 29–39, MR 0189915 .

External links

Wikimedia Commons has media related to Shannon multigraphs.

Lutz Volkmann: Graphen an allen Ecken und Kanten. Lecture notes 2006, p. 242 (German)

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