Graphs and matrices: A translation of "Graphok és matrixok" by Dénes Kőnig (1931)
aa r X i v : . [ m a t h . HO ] S e p Graphs and matrices: A translationof “Graphok és matrixok” by Dénes Kőnig (1931) ∗ Gábor Szárnyas [email protected]
Abstract
This paper, originally written in Hungarian by Dénes Kőnig in 1931,proves that in a bipartite graph, the minimum vertex cover and the max-imum matching have the same size. This statement is now known asKőnig’s theorem. The paper also discusses the connection of graphs andmatrices, then makes some observations about the combinatorial proper-ties of the latter.
Let G be a (finite) bipartite graph. This means that all closed paths in G are of even length, in other words, the vertices of G can be partitioned into twosets Π and Π such that all edges in G connect a vertex in Π with a vertexin Π . Let M be the maximal number of the edges in G which do not have acommon vertex. If vertices A , A , . . . , A v in G are such that all edges in G are incident to one of these vertices, we say that A , A , . . . , A v cover the edgesof G . We prove that the edges of G can be covered with M vertices .Let K = ( P Q , P Q , . . . , P M Q M ) be the set of M edges such that, in accordance with the definition of M , thevertices P i , Q i ( M = 1 , , . . . , M ) are distinct. Let vertices P i belong to set Π and vertices Q i to Π , and let Π ′ = ( P , P , . . . , P M ) , Π ′ = ( Q , Q , . . . , Q M ) , ∗ The talk “Graphs and matrices” was given at the seminar of the Lorand Eötvös Mathemat-ics and Physics Society (Eötvös Loránd Matematikai és Fizikai Társulat) on March 26, 1931.The paper accompanying the talk originally appeared in the “Matematikai és Fizikai Lapok”(Mathematical and Physical Journal), volume 38, 1931. The original paper in Hungarian isavailable online at http://real-j.mtak.hu/7307/ . This translation has been published withpermission by the János Bolyai Mathematical Society (Bolyai János Matematikai Társulat).Thanks to Anna Gujgiczer and Naomi Arnold for providing feedback on the initial drafts ofthe translation. Such a set of edges is now called an independent edge set or a matching . Instead of cover (“lefogják”/“lefedik”), the original text used the term exhaust (“kimerítik”).
1n a way that Π ′ is a subset of Π and Π ′ is a subset of Π . We base our proofon the notion of a « K -path ».A K -path in G is a path (which is open and does not have repeating ver-tices) A A . . . A r in which the second, fourth, . . . , v th , . . . , and penultimateedge, i.e. the edges A A , A A , . . . , A v A v +1 , . . . , A r − A r − all belong to K .First, we prove the following lemma: There is no K -path in G that connects a vertex in Π − Π ′ to anothervertex in Π − Π ′ . Suppose U would be such a path, then by removing the edges of U ∩ K from K and adding the edges of U \ K (where the size of the latter set is greater by1), we would obtain M + 1 edges which do not share a vertex. This contradictsthe maximal nature of M .Now we define a subset of Π ′ + Π ′ , Π ′ = ( R , R , . . . , R M ) : let α be any of , , . . . , M and let R α = Q α if some K -path connects a vertex in Π − Π ′ with Q α ; if there is no such K -path, let R α = P α . This way, Π ′ contains an endpointof all the edges in K . We prove that the set Π ′ of M vertices covers the edgesof G , i.e. – given that P Q is an arbitrary edge of G (where P is in Π and Q isin Π ) – either P or Q is in Π ′ . Our proof distinguishes between four cases: Case 1.
Let P belong to Π − Π ′ and Q to Π − Π ′ . By adding this P Q edge to K , we would obtain M + 1 edges which do not share a vertex. Thiscontradicts the maximal property of M , therefore, this case is not possible. Case 2.
Let P belong to Π − Π ′ and Q to Π ′ . Then, Q = Q α , where α = 1 , , . . . , or M and edge P Q alone forms a K -path which connects vertex P in Π − Π ′ to Q = Q α . Therefore, Q = Q α belongs to Π ′ . Case 3.
Let P belong to Π ′ and Q to Π − Π ′ . Then, P = P α , where α = 1 , , . . . , or M . If there were a K -path which connects some vertex P of Π − Π ′ with Q α , then by adding edges Q α P α and P α Q we would derive a K -path, which connects P with Q . However, this is impossible according toour lemma. Therefore, there is no K -path which connects a vertex in Π − Π ′ to Q α . Therefore, P = P α belongs to Π ′ . Case 4.
Let P belong to Π ′ and Q to Π ′ . Let e.g. P = P α , Q = Q β . If α = β , then trivially either P = P α or Q = Q α belong to Π ′ . Therefore, assumethat α = β . Either P = P α belongs to Π ′ or there exists a K -path, whichconnects some vertex P of Π − Π ′ with Q α ; in the latter case, by adding edges Q α P α and P α Q β to this K -path, we derive a K -path which connects P with Q β in way that Q = Q β belongs to Π ′ .With this, we have indeed proved that if a bipartite graph maximally has M edges which do not have a common vertex, then the edges of G can be coveredby M vertices. Therefore, if m is the minimal number of vertices which coverthe vertices of G , then m ≦ M .It is obvious that the opposite is also true: m ≧ M . If viz. e , e , . . . , e M are edges which do not have a common vertex and vertices A , A , . . . , A m cover the edges of the graph, then all of edges e , e , . . . , e M end in one of thevertices A , A , . . . , A m ; and as these do not have any common endpoints, indeed m ≧ M . 2ith this, we have proved that m = M . To summarize, our main result canbe stated as follows: In a bipartite graph, the minimal number of vertices covering alledges is equal to the maximal number of edges which do not have acommon vertex.
Turning our attention to the application of this theorem on matrices, let k a ik k ( i = 1 , , . . . , p ; k = 1 , , . . . , q ) be any matrix where regarding the value of a certain element, we only considerwhether it “vanishes” or not. This matrix corresponds to a bipartite graph asfollows. Each of the p rows corresponds to one of the vertices P , P , . . . , P p ,each of the q columns corresponds to one of vertices Q , Q , . . . , Q q ; furthermorewe create a P i Q k edge iff the corresponding a ik element does not vanish. We donot create any other edges. This way, we construct a bipartite graph G .The notion that vertices cover the edges of G clearly means that the set ofvertices corresponding to these rows and columns (in general: lines ) contain allnon-vanishing elements of the matrix. Meanwhile, the notion that certain edgesdo not have a common vertex means that the elements corresponding to theseedges do not lie on the same line.Overall, our result for matrices can be summarized as follows: For any matrix, the minimal number of lines which contain all non-vanishing elements is equal to the maximal number of non-vanishingelements which pairwise do not lie on the same line.
It is obvious that the term «non-vanishing» can be substituted with anyproperty of the elements, therefore this theorem expresses a purely combinatorialproperty of matrices (two-dimensional tables) where elements can be any objects(not just numbers).Finally, we mention that our results are closely related to the research ondeterminants by
Frobenius and on graphs by
Menger . We will discuss theseconnections later.
Dénes Kőnig. The term vanishes is a literal translation of the word “eltűnik” (meaning vanishes, disap-pears, or fades away). In recent works on sparse linear algebra, these elements are called zeroelements , while non-vanishing elements are called non-zero elements ..