Shannon multigraph

In the mathematical discipline of graph theory, Shannon multigraphs, named after Claude Shannon by, 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 $$\left\lfloor \frac{n+1}{2} \right\rfloor $$.

Edge coloring
According to a theorem of, 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 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.