Positivity of three-term recurrence sequences.
In [1], Brousek characterizes all triples of connected graphs, G₁,G₂,G₃, with for some i = 1,2, or 3, such that all G₁G₂ G₃-free graphs contain a hamiltonian cycle. In [8], Faudree, Gould, Jacobson and Lesniak consider the problem of finding triples of graphs G₁,G₂,G₃, none of which is a , s ≥ 3 such that G₁G₂G₃-free graphs of sufficiently large order contain a hamiltonian cycle. In [6], a characterization was given of all triples G₁,G₂,G₃ with none being , such that all G₁G₂G₃-free graphs are...
We extend the notion of a potentially H-graphic sequence as follows. Let A and B be nonnegative integer sequences. The sequence pair S = (A,B) is said to be bigraphic if there is some bipartite graph G = (X ∪ Y,E) such that A and B are the degrees of the vertices in X and Y, respectively. If S is a bigraphic pair, let σ(S) denote the sum of the terms in A. Given a bigraphic pair S, and a fixed bipartite graph H, we say that S is potentially H-bigraphic if there is some realization of S containing...
The set of all non-increasing nonnegative integer sequences () is denoted by . A sequence is said to be graphic if it is the degree sequence of a simple graph on vertices, and such a graph is called a realization of . The set of all graphic sequences in is denoted by . A graphical sequence is potentially -graphical if there is a realization of containing as a subgraph, while is forcibly -graphical if every realization of contains as a subgraph. Let denote a complete...
The power index of a square Boolean matrix A is the least integer d such that Ad is a linear combination of previous nonnegative powers of A. We determine the maximum power indices for the class of n×n primitive symmetric Boolean matrices of trace zero, the class of n×n irreducible nonprimitive symmetric Boolean matrices, and the class of n×n reducible symmetric Boolean matrices of trace zero, and characterize the extreme matrices respectively.
We prove that every planar graph with maximum degree ∆ is strong edge (2∆−1)-colorable if its girth is at least 40 [...] +1. The bound 2∆−1 is reached at any graph that has two adjacent vertices of degree ∆.
Let be a field and . Let be a monomial ideal of and be monomials in . We prove that if form a filter-regular sequence on , then is pretty clean if and only if is pretty clean. Also, we show that if form a filter-regular sequence on , then Stanley’s conjecture is true for if and only if it is true for . Finally, we prove that if is a minimal set of generators for which form either a -sequence, proper sequence or strong -sequence (with respect to the reverse lexicographic...
We consider a special packing-covering pair of problems. The packing problem is a natural generalization of finding a (weighted) maximum independent set in an interval graph, the covering problem generalizes the problem of finding a (weighted) minimum clique cover in an interval graph. The problem pair involves weights and capacities; we consider the case of unit weights and the case of unit capacities. In each case we describe a simple algorithm that outputs a solution to the packing problem and...