Kőnig's theorem (graph theory)

Setting

A graph is bipartite if its vertices can be partitioned into two sets such that each edge has one endpoint in each set. A vertex cover in a graph is a set of vertices that includes at least one endpoint of every edge, and a vertex cover is minimum if no other vertex cover has fewer vertices. A matching in a graph is a set of edges no two of which share an endpoint, and a matching is maximum if no other matching has more edges. König's theorem states that, in any bipartite graph, the number of edges in a maximum matching is equal to the number of vertices in a minimum vertex cover.

For graphs that are not bipartite, the maximum matching and minimum vertex cover problems are very different in complexity: maximum matchings can be found in polynomial time for any graph, while minimum vertex cover is NP-complete. The complement of a vertex cover in any graph is an independent set, so a minimum vertex cover is complementary to a maximum independent set; finding maximum independent sets is another NP-complete problem. The equivalence between matching and covering articulated in König's theorem allows minimum vertex covers and maximum independent sets to be computed in polynomial time for bipartite graphs, despite the NP-completeness of these problems for more general graph families.

König's theorem is equivalent to numerous other min-max theorems in graph theory and combinatorics, such as Hall's marriage theorem and Dilworth's theorem. Since bipartite matching is a special case of maximum flow, the theorem also results from the max-flow min-cut theorem.

History

König's theorem is named after the Hungarian mathematician Dénes Kőnig. Kőnig had announced in 1914 and published in 1916 the results that every regular bipartite graph has a perfect matching, and more generally that the chromatic index of any bipartite graph (that is, the minimum number of matchings into which it can be partitioned) equals its maximum degree – the latter statement is known as König's Line Coloring Theorem. However, Bondy & Murty (1976) attribute König's theorem itself to a later paper of Kőnig (1931). According to Biggs, Lloyd & Wilson (1976), Kőnig attributed the idea of studying matchings in bipartite graphs to his father, mathematician Gyula Kőnig. In Hungarian, Kőnig's name has a double acute accent, but his theorem is usually spelled in German characters, with an umlaut.

Statement of the theorem

In any bipartite graph, the number of edges in a maximum matching equals the number of vertices in a minimum vertex cover.

Example

The bipartite graph shown in the above illustration has 14 vertices; a matching with six edges is shown in blue, and a vertex cover with six vertices is shown in red. There can be no smaller vertex cover, because any vertex cover has to include at least one endpoint of each matched edge (as well as of every other edge), so this is a minimum vertex cover. Similarly, there can be no larger matching, because any matched edge has to include at least one endpoint in the vertex cover, so this is a maximum matching. König's theorem states that the equality between the sizes of the matching and the cover (in this example, both numbers are six) applies more generally to any bipartite graph.

Proof

König's theorem can be proven in a way that provides additional useful information beyond just its truth: the proof provides a way of constructing a minimum vertex cover from a maximum matching. Let G = ( V , E ) be a bipartite graph, and let the vertex set V be partitioned into left set L and right set R . Suppose that M is a maximum matching for G . No vertex in a vertex cover can cover more than one edge of M (because the edge half-overlap would prevent M from being a matching in the first place), so if a vertex cover with | M | vertices can be constructed, it must be a minimum cover.

To construct such a cover, let U be the set of unmatched vertices in L (possibly empty), and let Z be the set of vertices that are either in U or are connected to U by alternating paths (paths that alternate between edges that are in the matching and edges that are not in the matching). Let

Every edge e in E either belongs to an alternating path (and has a right endpoint in K ), or it has a left endpoint in K . For, if e is matched but not in an alternating path, then its left endpoint cannot be in an alternating path (for such a path could only end at e ) and thus belongs to L ∖ Z . Alternatively, if e is unmatched but not in an alternating path, then its left endpoint cannot be in an alternating path, for such a path could be extended by adding e to it. Thus, K forms a vertex cover.

Additionally, every vertex in K is an endpoint of a matched edge. For, every vertex in L ∖ Z is matched because Z is a superset of U, the set of unmatched left vertices. And every vertex in R ∩ Z must also be matched, for if there existed an alternating path to an unmatched vertex then changing the matching by removing the matched edges from this path and adding the unmatched edges in their place would increase the size of the matching. However, no matched edge can have both of its endpoints in K . Thus, K is a vertex cover of cardinality equal to M , and must be a minimum vertex cover.

Algorithm

The construction described in the proof above provides an algorithm for producing a minimum vertex cover given a maximum matching. Thus, the Hopcroft–Karp algorithm for finding maximum matchings in bipartite graphs may also be used to solve the vertex cover problem efficiently in these graphs.

Despite the equivalence of the two problems from the point of view of exact solutions, they are not equivalent for approximation algorithms. Bipartite maximum matchings can be approximated arbitrarily accurately in constant time by distributed algorithms; in contrast, approximating the minimum vertex cover of a bipartite graph requires at least logarithmic time.

Connections with perfect graphs

A graph is said to be perfect if, in every induced subgraph, the chromatic number equals the size of the largest clique. Any bipartite graph is perfect, because each of its subgraphs is either bipartite or independent; in a bipartite graph that is not independent the chromatic number and the size of the largest clique are both two while in an independent set the chromatic number and clique number are both one.

A graph is perfect if and only if its complement is perfect, and König's theorem can be seen as equivalent to the statement that the complement of a bipartite graph is perfect. For, each color class in a coloring of the complement of a bipartite graph is of size at most 2 and the classes of size 2 form a matching, a clique in the complement of a graph G is an independent set in G, and as we have already described an independent set in a bipartite graph G is a complement of a vertex cover in G. Thus, any matching M in a bipartite graph G with n vertices corresponds to a coloring of the complement of G with n-|M| colors, which by the perfection of complements of bipartite graphs corresponds to an independent set in G with n-|M| vertices, which corresponds to a vertex cover of G with M vertices. Conversely, König's theorem proves the perfection of the complements of bipartite graphs, a result proven in a more explicit form by Gallai (1958).

One can also connect König's Line Coloring Theorem to a different class of perfect graphs, the line graphs of bipartite graphs. If G is a graph, the line graph L(G) has a vertex for each edge of G, and an edge for each pair of adjacent edges in G. Thus, the chromatic number of L(G) equals the chromatic index of G. If G is bipartite, the cliques in L(G) are exactly the sets of edges in G sharing a common endpoint. Now König's Line Coloring Theorem, stating that the chromatic index equals the maximum vertex degree in any bipartite graph, can be interpreted as stating that the line graph of a bipartite graph is perfect.

Since line graphs of bipartite graphs are perfect, the complements of line graphs of bipartite graphs are also perfect. A clique in the complement of the line graph of G is just a matching in G. And a coloring in the complement of the line graph of G, when G is bipartite, is a partition of the edges of G into subsets of edges sharing a common endpoint; the endpoints shared by each of these subsets form a vertex cover for G. Therefore, König's theorem itself can also be interpreted as stating that the complements of line graphs of bipartite graphs are perfect.