A graph G is called a fractional (k, n′,m)-critical deleted graph if any n′ vertices are removed from G the resulting graph is a fractional (k,m)-deleted graph. In this paper, we prove that for integers k ≥ 2, n′,m ≥ 0, n ≥ 8k + n′ + 4m− 7, and δ(G) ≥ k + n′ +m, if |NG(x) ∪NG(y)| ≥ n+ n′ 2 for each pair of non-adjacent vertices x, y of G, then G is a fractional (k, n′,m)-critical deleted graph....

Matematiska Institutionen, Lunds Universitet, SE-221 00 Lund, Sweden. E-mail: [email protected], [email protected] Institut für Mathematik der Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria. E-mail: [email protected] WWW: http://www.mat.univie.ac.at/~kratt Abstract. We show how to find arbitrarily fast convergent series expansions for π of the form π = ∑∞ n=0 S(n)...

A topological fact about eleven dimensions is used to motivate a potential new duality in M-theory. We complete the discussion of consistent limits of M-theory raised in a previous paper [1], to include gravitational anomaly cancelation and four-form flux quantization in the context of the M × S1/Z2 compactifications. A suprise is found: If one includes a Z2 anomaly which exists only in 8k + 3 ...

We prove a decomposition result for locally finite graphs which can be used to extend results on edge-connectivity from finite to infinite graphs. It implies that every 4k-edge-connected graph G contains an immersion of some finite 2k-edge-connected Eulerian graph containing any prescribed vertex set (while planar graphs show that G need not contain a subdivision of a simple finite graph of lar...

This paper considers a degree sum condition sufficient to imply the existence of k vertexdisjoint cycles in a graph G. For an integer t ≥ 1, let σt (G) be the smallest sum of degrees of t independent vertices of G. We prove that if G has order at least 7k+1 and σ4(G) ≥ 8k−3, with k ≥ 2, then G contains k vertex-disjoint cycles. We also show that the degree sum condition on σ4(G) is sharp and co...

A graph G is k-crossing-critical if cr(G)≥k, but cr(G∖e)<k for each edge e∈E(G), where cr(G) the crossing number of G. It known that latest upper bound a 2k+8k+47 when δ(G)≥3, and 2k+35 δ(G)≥4, δ(G) minimum degree In this paper, we mainly show any with n vertices, cr(G)≤2k+8 cr(G)≤2k−k/2n+35/6 δ(G)≥5.

We prove the following best possible result. Let k ≥ 2 be an integer and G be a graph of order n with minimum degree at least k. Assume n ≥ 8k− 16 for even n and n ≥ 6k−13 for odd n. If the degree sum of each pair of nonadjacent vertices of G is at least n, then for any given Hamiltonian cycle C of G, G has a [k, k + 1]-factor containing C. Submitted: December 15, 1997; Accepted: November 27, 1...