Advances in Graph Theory by B. Bollobás (Eds.) PDF

By B. Bollobás (Eds.)

ISBN-10: 0720408431

ISBN-13: 9780720408430

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If G, and G, can be decomposed into hamiltonian cycles, then G, . G, also can be decomposed into hamiltonian cycles. Proof. This follows from the distributivity of the product. with respect to the edge disjoint union of graphs (a property not holding for the Cartesian sum and the lexicographic product). 20. Many other problems similar to those above can be considered; in particular we can consider decompositions into cycles of given length (see [6]). We mention also that Huang and Rosa [9] have considered “orthogonal” hamiltonian decompositions, and finally we give the following conjecture of Kotzig [11].

Hence there exists an x, E L , and a W, c Wl, I W,l 2 a2n,such that x, is not joined to W, at all. If x, does not represent all the L,'s, we may assume that x, FZ L2. Iterating this argument we find an X'E L, not joined to a W3 c W, at all, where 1 W,l a u3n, and if x,, x2 do not represent all the Lp's,we define xj and W, in the same way. Generally, if x p and W,+, are already defined, we check whether the set X , = {x,, . . , x p } represents all the L c A,. Bollobas, P. Simonovits, E. Szemeridi so that xPcl is not joined to W,, at all.

For each positive odd integer n, the edges of the complete graph K,, can be partitioned into n sets each being a matching with ( n - 1)/2 edges. Let H be a subgroup of the symmetric group S, with index t. Let H , = H , H 2 , . . , H , be an enumeration of the right cosets of S,, with respect to H. Let 51 Chromatic index of the graph of the assignment polytope G , ( H i ) denote the subgraph of G, induced by the vertices in Hi(1s i c t). For distinct integers i and j with 1 s i, j c t, let Gn(Hi,Hi) denote the partial subgraph of G,, where the set of vertices is Hi U Hi and where two vertices are joined by an edge if and only if one is in Hi, the other is in Hi, and they are joined by an edge in G,.

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Advances in Graph Theory by B. Bollobás (Eds.)

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