Shannon multigraph

In the mathematical discipline of graph theory, Shannon multigraphs, named after Claude Shannon by Vizing (1965), ^[1] 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 ${\displaystyle \left\lfloor {\frac {n}{2}}\right\rfloor }$, ${\displaystyle \left\lfloor {\frac {n}{2}}\right\rfloor }$ and ${\displaystyle \left\lfloor {\frac {n+1}{2}}\right\rfloor }$ 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 ${\displaystyle \left\lfloor {\frac {n+1}{2}}\right\rfloor }$. ^{[2] [3]}

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 ${\displaystyle \Delta }$ has an edge coloring that uses at most ${\displaystyle {\frac {3}{2}}\Delta }$ colors. ^[4] When ${\displaystyle \Delta }$ is even, the example of the Shannon multigraph with multiplicity ${\displaystyle \Delta /2}$ shows that this bound is tight: the vertex degree is exactly ${\displaystyle \Delta }$, but each of the ${\displaystyle {\frac {3}{2}}\Delta }$ edges is adjacent to every other edge, so it requires ${\displaystyle {\frac {3}{2}}\Delta }$ colors in any proper edge coloring. ^[3]

A version of Vizing's theorem states that every multigraph with maximum degree ${\displaystyle \Delta }$ and multiplicity ${\displaystyle \mu }$ may be colored using at most ${\displaystyle \Delta +\mu }$ colors. ^[5] Again, this bound is tight for the Shannon multigraphs. ^[3]

References

1. Vizing, V. G. (1965), "The chromatic class of a multigraph", Kibernetika, 1965 (3): 29–39, MR   0189915
2. Fiorini, S.; Wilson, Robin James (1977), Edge-colourings of graphs, Research Notes in Mathematics, vol. 16, London: Pitman, p. 34, ISBN   0-273-01129-4, MR   0543798
3. Volkmann, Lutz (1996), Fundamente der Graphentheorie (in German), Wien: Springer, p. 289, ISBN   3-211-82774-9
4. Shannon, Claude E. (1949), "A theorem on coloring the lines of a network", J. Math. Physics, 28: 148–151, doi:10.1002/sapm1949281148, hdl: 10338.dmlcz/101098 , MR   0030203
5. Vizing, V. G. (1964), "On an estimate of the chromatic class of a p-graph", Diskret. Analiz., 3: 25–30, MR   0180505

External links

• Lutz Volkmann: Graphen an allen Ecken und Kanten . Lecture notes 2006, p. 244 (German)