A recurrent 2-dimensional (double) sequence t(m,n) is given by fixing particular sequences t(m, 0), t(0, n) as initial conditions and a rule of recurrence t(m,n) = t(a(m,n − 1), t(m − 1, n − 1), t(m − 1, n)) for m,n ≥ 1. We display such a sequence with values in the set {0, 1} and show that all rows are periodic and that the minimal period of the n-th row has length 2. We conclude that this 2-d...

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A (k, l)-kernel N of D is a k-independent set of vertices (if u, v ∈ N then d(u, v), d(v, u) ≥ k) and l-absorbent (if u ∈ V (D) − N then there exists v ∈ N such that d(u, v) ≤ l). A k-kernel is a (k, k − 1)-kernel. A digraph D is cyclically k-partite if there exists a partition {Vi} i=0 of V (D) suc...

An optimization problem arising in the design of optical networks is shown here to be abstracted by the following model of covering the edges of a bipartite graph with a minimum number of 4-cycles, or K2,2: Given a bipartite graph G = (L,R,E) over the node set L ∪ R with E ⊆ {[u, v] : u ∈ L, v ∈ R}, and the implicit collection of all fournode cycles in the complete bipartite graph over L∪R. The...

Received Let G = (V; E) be a graph with n vertices, chromatic number (G) and list chromatic number`(G): Suppose each vertex of V (G) is assigned a list of t colors. Albertson, Grossman and Haas 1] conjectured that at least t ` (G) n vertices can be colored properly from these lists. Albertson et. al. 1] and Chappell 3] proved partial results concerning this conjecture. This paper presents algor...

Let q be a prime power, Fq be the field of q elements, and k;m be positive integers. A bipartite graph G 1⁄4 Gqðk;mÞ is defined as follows. The vertex set of G is a union of two copies P and L of two-dimensional vector spaces over Fq, with two vertices ðp1;p2Þ 2 P and 1⁄2 l1; l2 2 L being adjacent if and only if p2 þ l2 1⁄4 p 1 l 1 . We prove that graphs Gqðk ;mÞ and Gq0 ðk 0;m0Þ are isomorphic...

Let $G$ be a bipartite graph. In this paper we consider the two kind of location problems namely $p$-center and $p$-median problems on bipartite graphs. The $p$-center and $p$-median problems asks to find a subset of vertices of cardinality $p$, so that respectively the maximum and sum of the distances from this set to all other vertices in $G$ is minimized. For each case we present some proper...

A p-list assignment L of a graph G assigns to each vertex v of G a set L(v) ⊆ {0, 1, . . . , p − 1} of permissible colors. We say G is L-(p, q)-colorable if G has a (p, q)-coloring h such that h(v) ∈ L(v) for each vertex v. The circular list chromatic number χc,l(G) of a graph G is the infimum of those real numbers t for which the following holds: For any p, q, for any p-list assignment L with ...

Let G = (V,E) be a k-edge-connected graph with edge costs {c(e) : e ∈ E} and let 1 ≤ l ≤ k − 1. We show by a simple and short proof, that G contains an l-edge cover I such that: c(I) ≤ l k c(E) if G is bipartite, or if l|V | is even, or if |E| ≥ k|V | 2 + k 2l ; otherwise, c(I) ≤ (

In this note we put forward a conjecture on the average optimal length for bipartite matching with a finite number of elements where the different lengths are independent one from the others and have an exponential distribution. The problem of random bipartite matching (or assignment) is interesting both from the point of view of optimisation theory and of statistical mechanics [1, 2, 3, 4]. Ea...