Vizing’s theorem

class number

Definitions and notation

In a graph or multigraph $G$, let $\rho (\mathrm{\text{v}})$ denote the valency of vertex(node) v, and let $\rho (G)$ denote the largest valency in $G$ (often writtenjust $\rho$ on its own, $\mathrm{\Delta }(G)$ is another common notation).

Let the multiplicity $\mu (\mathrm{\text{v}},\mathrm{\text{w}})$ of vertices (nodes) v and wbe the number of parallel edges that link them, and here too let $\mu (G)$ bethe largest multiplicity in $G$. A graph is a multigraph for which $\mu (G)=1$.

An edge coloring of a (multi)graph $G$ is a mapping from theset $E$ of its edges to a set $K$ of items called colors, in such a waythat at any vertex v, the $\rho (\mathrm{\text{v}})$ edges there all have a differentcolor. An edge-$k$-coloring is an edge-coloring where $|K|=k$.

Note that a loop (an edge joining v to itself) accounts for two of the$\rho (\mathrm{\text{v}})$ edges at v and cannot have a color different from itself,so edge-colorings as defined here do not exist for pseudographs (structuresthat are allowed to have loops).

The chromatic index (aka edge-chromatic number) ${\chi }^{\prime }(G)$ isthe smallest number $k$ for which an edge-$k$-coloring of $G$ exists. This nowstandard notation is analogous to the chromatic number $\chi (G)$ whichrefers to vertex coloring.

Vizing’s theorem

For any multigraph $G$ (this includes graphs), we have of course

$${\chi }^{\prime }(G)⩾\rho (G)$$

immediately from the definition. Surprisingly though, we also have

Theorem (Vizing)$$For any graph $G$,

$${\chi }^{\prime }(G)⩽\rho (G)+1$$

This celebrated theorem was proved by V. G. Vizing in 1964 while still agraduate student (at Novosibirsk). It is a special case of

Theorem (Vizing)$$For any multigraph $G$,

$${\chi }^{\prime }(G)⩽\rho (G)+\mu (G)$$

(edge-coloring of multigraphs is the subject of Vizing’s doctoral thesis, 1965).The theorem was proved independently by R. P. Gupta in 1966; nowadays variousversions of the proofs exist. A key rôle is played by connected subgraphs $H(a,b)$ of colors $a$ and $b$ that cannot be extended further. They are analogous to Kempe chains in face or vertex colorings, but have a simpler structure: they are either paths or closed paths (the latter of even length). http://planetmath.org/node/6932See here for a proof (in the case of graphs).

Ore [Ore67] gives a sharper bound for multigraphs. Let theenlarged valency ${\rho }^{+}(\mathrm{\text{v}})$ be given by

$${\rho }^{+}(\mathrm{\text{v}})\stackrel{\text{def}}{=}\rho (\mathrm{\text{v}})+\underset{\mathrm{w}}{\max }\mu (\mathrm{\text{v}},\mathrm{\text{w}})$$

where w can be taken over all vertices of the graph; it effectively onlyranges over those adjacent to v (as $\mu (\mathrm{\text{v}},\mathrm{\text{w}})$ is zero for the others).Again, let ${\rho }^{+}(G)$ denote the maximum enlarged valency occurring in$G$. Now

Theorem (Ore)$$ For any multigraph $G$,

$${\chi }^{\prime }(G)⩽{\rho }^{+}(G)$$

Shannon’s theorem

The following theorem was proven in 1949 byClaude E. Shannon. He also gives examples, for any value of $\rho$, ofmultigraphs that actually attain the bound,the so-called Shannon graphs: they have three vertices, with$\mu (\mathrm{\text{a}},\mathrm{\text{b}})=\mu (\mathrm{\text{a}},\mathrm{\text{c}})=\lfloor \rho /2\rfloor$ and$\mu (\mathrm{\text{b}},\mathrm{\text{c}})=\lceil \rho /2\rceil$.

Theorem (Shannon)$$ For any multigraph $G$,

$${\chi }^{\prime }(G)⩽\lfloor \frac{3}{2}\rho (G)\rfloor$$

While giving a much worse bound for graphs, it gives for some multigraphs abetter bound than Vizing’s theorem. Nevertheless, it is also possible to proveit from the latter [FW77].

Only in the context of graph colorings is Shannon’s theorem understoodto refer to the one here; in the wider world the term tends to refer to any ofhis fundamental theorems in information theory.

Here too Ore [Ore67] gives a sharper bound based on the maximum ofa local expression. Let $\sigma (\mathrm{\text{v}})$ be 0 if v has fewer than twoneighbours, and otherwise

$$\sigma (\mathrm{\text{v}})\stackrel{\text{def}}{=}\underset{{\mathrm{v}}^{\prime },{\mathrm{v}}^{\prime \prime }}{\max }(\rho (\mathrm{\text{v}})+\rho ({\mathrm{\text{v}}}^{\prime })+\rho ({\mathrm{\text{v}}}^{\prime \prime }))$$

where ${\mathrm{\text{v}}}^{\prime }$ and ${\mathrm{\text{v}}}^{\prime \prime }$ range over all pairs of distinct neighbours of v. Again, let $\sigma (G)$ be itslargest value in the graph.

Theorem (Ore)$$ For any multigraph $G$,

$${\chi }^{\prime }(G)⩽\max (\rho (G),\lfloor \frac{1}{2}\sigma (G)\rfloor )$$

Chromatic class

Vizing’s theorem has the effect of placing each multigraph $G$ in one of theclasses 0, 1, 2, … $\mu (G)$ where the class number is${\chi }^{\prime }(G)-\rho (G)$.

For graphs it means they split into just two classes: those that can beedge-colored in $\rho (G)$ colors and those that need $\rho (G)+1$colors. The logical name for them would be class 0 and class 1; unfortunatelythe standard terminology is class 1 and 2 (or I and II).

Class I (graphs that can be edge-$\rho$-colored) contains among others

• single $n$-cycles for even $n$ ${}^{†}$

• complete graphs $K_n$ for even $n$ ${}^{‡}$

• bipartite graphs (this is König’s theorem, 1916)

• bridgeless planar trivalent graphs (four-color theorem via Tait coloring)

• planar graphs with $\rho (G)⩾8$ (by another theorem of Vizing)

• planar graphs with $\rho (G)⩾6$?? (conjecture of Vizing)

Class II (graphs that need $\rho +1$ colors) contains among others

• single $n$-cycles for odd $n$ ${}^{†}$

• complete graphs $K_n$ for odd $n$ ${}^{‡}$

• $K_n$ for odd $n$ with a few edges missing (by a couple of theorems)

• trivalent graphs with a bridge ${}^{\mathrm{\P }}$

but the general classification problem has thus far eluded the best efforts ofVizing and many others. There are interesting links here with polyhedraldecompositions (aka cyclic double covers) and embeddings in surfaces.

A(n edge-)critical graph is a connected graph of class II but suchthat removing any of its edges makes it class I. As often in graph theory,such a minimality condition imposes a certain amount of structure on the graph.There are conjectures…

Almost all graphs are in class I

Let $\mathrm{\#}{G}_n$ be the number of graphs with $n$ vertices,and $\mathrm{\#}{G}_n^{\mathrm{I}}$ the number of them in class I.

Theorem (P. Erdős and R. J. Wilson)$$

$$\underset{n\to \mathrm{\infty }}{lim}\frac{\mathrm{\#}{G}_n^{\mathrm{I}}}{\mathrm{\#}{G}_n}=1$$

So graphs of class II get relatively rarer for larger graph sizes. The absolute numbers do still increase. For cubic graphs for instance, we saw every one with a bridge is in class II. Bridgeless cubic graphs of class II are a bit thinner on the ground. By the four-color theorem, via Tait coloring, we know all of them are non-planar. Rarer still are those of them with girth at least five (and some non-triviality conditions); they are so hard to find that Martin Gardner dubbed them snarks. The Petersen graph is one, and a few infinite families of snarks have been found.

$†\mathit{\hspace{1em}}$ $\rho =2$, so an edge-$\rho$-coloring must use alternating colors. For odd $n$ that’s impossible.

$‡\mathit{\hspace{1em}}$ $\rho =n-1$, note it is also the valency of every vertex.Fix two colors $a$ and $b$. A Kempe chain $H(a,b)$ can only terminate at a vertex where one of those colors is missing but in $K_n$ we cannot afford to miss any color at any vertex, so every $H(a,b)$ is a cycle of even length and together they visit all $n$ vertices. For odd $n$ that’s impossible.For even $n$ there are ways to construct the coloring (try it).

$\mathrm{\P }$ $\rho =3$ and again the valency of every vertex. For the samereason as in the previous note, every $H(a,b)$ or $H(c,a)$ or $H(b,c)$ is a cycle. Every edge must be in two such, but a bridge cannot be part of a cycle.