3 For d, k ∈ N with k ≤ 2d, let g(d, k) denote the infimum density of binary 4 sequences (xi)i∈Z ∈ {0, 1}Z which satisfy the minimum degree condition d ∑ j=1 (xi+j + 5 xi−j) ≥ k for all i ∈ Z with xi = 1. We reduce the problem to determine g(d, k) 6 to a combinatorial problem related to the generalized k-girth of a graph G which 7 is defined as the minimum order of an induced subgraph of G of m...

Given a bipartite graph H and a positive integer n such that v(H) divides 2n, we define the minimum degree threshold for bipartite H-tiling, δ2(n,H), as the smallest integer k such that every bipartite graph G with n vertices in each partition and minimum degree δ(G) ≥ k contains a spanning subgraph consisting of vertex-disjoint copies of H. Zhao, Hladký-Schacht, Czygrinow-DeBiasio determined δ...

Let fr(n, e) be the minimum number of r-cliques in graphs of order n and size e. Determining fr(n, e) has been a long-studied problem. The case r = 3, that is, counting triangles, has been studied by various people. Erdős [3], Lovász and Simonovits [7] studied the case when e = ( n 2 ) /2 + l with 0 < l n/2. Fisher [4] considered the situation when ( n 2 ) /2 e 2 ( n 2 ) /3, but it was not unti...

Total and average distance are not only interesting invariants of graphs in their own right but are also used for studying properties or classifying graphical systems that depend on the number of edges traversed. Recent examples include studies of computer networks [3] and the use of graphical invariants to partially classify the structure of molecules [1]. There have been a number of conjectur...

It is well known that every bipartite graph with vertex classes of size n whose minimum degree is at least n/2 contains a perfect matching. We prove an analogue of this result for hypergraphs. We also prove several related results which guarantee the existence of almost perfect matchings in r-uniform hypergraphs of large minimum degree. Our bounds on the minimum degree are essentially best poss...

For a k-linked graph G and a vector E S of 2k distinct vertices of G, an E S-linkage is a set of k vertex-disjoint paths joining particular vertices of E S. Let T denote theminimum order of an E S-linkage in G. A graph G is said to be pan-k-linked if it is k-linked and for all vectors E S of 2k distinct vertices of G, there exists an E S-linkage of order t for all t such that T ≤ t ≤ |V (G)|. W...

A classic result of Dirac states that if G is a 2-connected graph of order n with minimum degree δ ≥ 3, then G contains a cycle of length at least min{n, 2δ}. In this paper, we consider the problem of determining the number of different odd or even cycle lengths that must exist under the minimum degree condition. We conjecture that there are δ − 1 even cycles of different lengths, and when G is...

We consider the minimum spanning tree (MST) problem under the restriction that for every vertex v, the edges of the tree that are adjacent to v satisfy a given family of constraints. A famous example thereof is the classical degree-bounded MST problem, where for every vertex v, a simple upper bound on the degree is imposed. Iterative rounding/relaxation algorithms became the tool of choice for ...

The paper ‘‘Mean Distance and Minimum Degree,’’ by Mekkia Kouider and Peter Winkler, JGT 25#1 (1997), 95–99 mistakenly attributes the computer program GRAFFITI to Fajtlowitz and Waller, instead of just Fajtlowitz. (Our apologies to Siemion Fajtlowitz.) Note also that one of the ‘‘flaws’’ we note for Conjecture 62 (that it was made for graphs regular of degree d, vice graphs of minimum degree d)...

The problem of matrix inversion is central to many applications of Numerical Linear Algebra. When the matrix to invert is dense, little can be done to avoid the costly O(n) process of Gaussian Elimination. When the matrix is symmetric, one can use the Cholesky Factorization to reduce the work of inversion (still O(n), but with a smaller coefficient). When the matrix is both sparse and symmetric...