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Shannon multigraph

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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.

Contents

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 ⌊ n 2 ⌋ , ⌊ n 2 ⌋ and ⌊ n + 1 2 ⌋ 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 ⌊ n + 1 2 ⌋ .

    Examples

  • Shannon multigraphs

    • Edge coloring

    According to a theorem of Shannon (1949), every multigraph with maximum degree Δ has an edge coloring that uses at most 3 2 Δ colors. When Δ is even, the example of the Shannon multigraph with multiplicity Δ / 2 shows that this bound is tight: the vertex degree is exactly Δ , but each of the 3 2 Δ edges is adjacent to every other edge, so it requires 3 2 Δ colors in any proper edge coloring.

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

    References

    Shannon multigraph Wikipedia
    (Text) CC BY-SA


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