D. K onig, " Uber graphen und ihre anwendung auf determinantentheorie und mengenlehre," Math. Ann., vol. 77, pp. 453--465, 1916.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....each input switch I and at most one request to each output switch J . If m is the maximum number of requests involving an input switch or an output switch, then m middle stage switches is necessary and sufficient to route all requests. This is a consequence of the Konig s Line Coloring Theorem [9]. For another proof using P. Hall s matching condition, see [1] For example, when there are at most n requests out of each input switch or to each output switch, n middle stage switches is sufficient. This is the celebrated Slepian Duguid theorem [1] Consequently, in order to determine how many ....

D. K ONIG, Uber graphen und ihre anwendung auf determinantentheorie und mengenlehre, Math. Ann., 77 (1916), pp. 453--465.

....Notice that when all the weights are 1, this problem reduces to the edge coloring of a bipartite graph with maximum degree at most n. Thus, m(n; r) n when the weights are all unity. This can be shown as a trivial consequence of P. Hall s matching condition, of Konig s Line Coloring Theorem [9]. 3 A new lower bound Theorem 3.1. For integers n; r 2, we have m(n; r) m(n; 2) Furthermore, when n is even; and 5n 1 ; when n is odd: Proof. The natural approach to find a lower bound k for m(n; r) is to find a particular graph G 2 B r which requires at least k colors. The ....

....2 A of H . We have deg H (a) minfjL a (b)j; jL b (a)jg L a (b) n; 21) by (19) Similarly, deg H (b) n for all b 2 B. Add more edges of G into H so that H is n regular. This is possible since G has n parallel edges between any pair (a; b) 2 A B. Konig s Line Coloring Theorem [9] implies that H is n edge colorable. The graph G is (r 1)n regular, hence it is (r 1)n edge colorable. However, each edge of G has weight at most 1=2, hence every two colors can be combined into one without violating the condition that the total weight of same color edges at each vertex is ....

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D. K ONIG, Uber graphen und ihre anwendung auf determinantentheorie und mengenlehre, Math. Ann., 77 (1916), pp. 453--465.

....the stable set polytopes of small imperfect graphs yields that G and G are the only two not weakly rank perfect graphs up to seven nodes. 3 Critical Edges in Line Graphs of Bipartite Graphs Let xy stand for a critical edge in the line graph G of a bipartite graph F . Then G is perfect by K onig [6] but G xy not anymore. We like to know whether removing only one critical edge leaves the graph still almost perfect or makes it already very imperfect . In order to decide how imperfect G xy is we formulate the following problem. 7 4 13 7 6 5 12 9 8 3 2 11 1 10 14 6 14 13 12 10 7 3 2 1 11 ....

D. Konig,  Uber Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Math. Ann. 77 (1916) 453-465

....degree. For each class, we give a lower bound on the size of matchings, and prove that it is tight for some graph within the class. 1 Introduction The problem of nding a maximum matching in a graph has a long and distinguished history beginning with the early work of Petersen [11] K onig [9], Hall [6] and Tutte [13] The fastest algorithms to nd a maximum matching in an n vertex m edge graph takes O( p nm) time, for bipartite graphs [7] as well as for general graphs [10] One intensely studied topic is whether a graph has a perfect matching, i.e. a matching of size n=2. This ....

....an n vertex m edge graph takes O( p nm) time, for bipartite graphs [7] as well as for general graphs [10] One intensely studied topic is whether a graph has a perfect matching, i.e. a matching of size n=2. This was shown for 3 regular biconnected graphs [11] and for k regular bipartite graphs [9], and the perfect matching can be found eciently for these graphs [2, 12, 4] Tutte [13] characterized when a graph has a perfect matching, but no algorithm that can nd a perfect matching in an arbitrary graph faster than nding a maximum matching is known. Not as much is known about bounds for ....

D. Konig.  Uber Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Mathematische Annalen, 77:453-465, 1916.

....graphs and multigraphs. The chromatic index # # (M) of a multigraph M is defined to be the smallest number k such that the edges of M can be colored with k colors where adjacent edges are colored di#erently. A trivial lower bound for the chromatic index of M is the maximum degree #(M) of M . Konig [21] proved that this lower bound is attained for bipartite multigraphs. This was one of the first results determining the chromatic index for a class of multigraphs. The first upper bound for the chromatic index involving all multigraphs was obtained by Shannon [27] who proved that # # (M) # 3 2 ....

D. K onig, Uber Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre, Math. Ann. 77 (1916), 453--465

....3.2 Let G be perfect and a linegraph. An edge of G is critical iff it is H critical. Theorem 3.3 Let G be the linegraph of F . G is critically perfect iff F is a bipartite H graph. Proof. If) Let F be a bipartite H graph, then its linegraph G is perfect by K onig s Edge Colouring Theorem [11]; every edge of G is critical by Theorem 3.1. 3 (Only if) Let G be critically perfect and the linegraph of F . We know by Theorem 3.1 that the edges of F corresponding to adjacent nodes in G form an H pair. Thus, F is an H graph. We have to show that F is bipartite. Assume F contains an odd ....

D. K onig, Uber Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Math. Ann. 77 (1916) 453-465

....trees as a black box) and extends to matroids. 2 Theorem 2 has two important special cases. First, if G is bipartite with colour classes S and T , then an S T connector is nothing but an edge cover of G (a set of edges covering all vertices) and Theorem 2 specializes to a theorem of Konig [5] and Gupta [3] saying that the maximum number of edge disjoint edge covers in a bipartite graph is equal to the minimum vertex degree. Second, if either S or T is a singleton, then an S T connector is a set of edges containing a spanning tree of G, and Theorem 2 specializes to Lemma 1. Note ....

D. K onig, Uber Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre, Math. Ann. 77 (1916), 453--465.

....G (v)j: 3.7) Two of the first and most basic results in matching theory (see [7] are Theorem H. Hall s Marriage Theorem, 6] 7] A bipartite graph G = V 1 ; V 2 ; E) has a matching of V 1 into V 2 iff j Gamma(S)j S for all S ae V 1 (3.8) and Theorem K. K onig s Minimax Theorem [7] [10]) For every bipartite graph G (G) G) These theorems can easily be derived from each other. We need here a consequence of Theorem K. Corollary 1. If G = V 1 ; V 2 ; E) satisfies for two numbers dV1 , dV2 and for i = 1; 2 d G (v) dV i for all v 2 V i ; then (G) G) min i=1;2 ....

D. K onig, " Uber Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre", Math. Annalen 77, 453--465, 1916.

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D. K onig, " Uber graphen und ihre anwendung auf determinantentheorie und mengenlehre," Math. Ann., vol. 77, pp. 453--465, 1916.

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D. Konig,  Uber Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre, Math. Ann. 77 (1916), 453-465.

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D. Konig.  Uber Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Mathematische Annalen, 77, 453-465, 1916.

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