A Skolem sequence is a sequence a1, a2, . . . , a2n (where ai ∈ A = {1, . . . , n}), each ai occurs exactly twice in the sequence and the two occurrences are exactly ai positions apart. A set A that can be used to construct Skolem sequences is called a Skolem set. The existence question of deciding which sets of the form A = {1, . . . , n} are Skolem sets was solved by Thoralf Skolem [6] in 195...

An induced packing of odd cycles in a graph is a packing such that there is no edge in the graph between any two odd cycles in the packing. We prove that the problem is solvable in time O(n ) when the input graph is planar. We also show that deciding if a graph has an induced packing of two odd cycles is NP-complete.

An injective map f : E(G) → {±1, ±2, · · · , ±q} is said to be an edge pair sum labeling of a graph G(p, q) if the induced vertex function f*: V (G) → Z − {0} defined by f*(v) = (Sigma e∈Ev) f (e) is one-one, where Ev denotes the set of edges in G that are incident with a vetex v and f*(V (G)) is either of the form {±k1, ±k2, · · · , ±kp/2} or {±k1, ±k2, · · · , ±k(p−1)/2} U {k(p+1)/2} accordin...

It was shown by Abrham that the number of pure Skolem sequences of order n, n ≡ 0 or 1 (mod 4), and the number of extended Skolem sequences of order n, are both bounded below by 2bn/3c. These results are extended to give similar lower bounds for the numbers of hooked Skolem sequences, split Skolem sequences and split-hooked Skolem sequences.

The identification of diametrical vertices in the d-dimensional hypercube (d >/3) leads to a (0, 2)-graph of degree d on 2 d1 vertices and of diameter [. d/2 J namely the extended odd graph (or Laborde-Mulder graph) for odd values of d, and the half-cube for even values of d. In this paper we prove that the diameter of a (0, 2)-graph of degree d on 2 ~~ vertices is at least [. d/2 .~ and when d...

Let G be a (p,q) graph. An injective map f : E(G) → {±1,±2,...,±q} is said to be an edge pair sum labeling if the induced vertex function f*: V (G) → Z - {0} defined by f*(v) = ΣP∈Ev f (e) is one-one where Ev denotes the set of edges in G that are incident with a vertex v and f*(V (G)) is either of the form {±k1,±k2,...,±kp/2} or {±k1,±k2,...,±k(p-1)/2} U {±k(p+1)/2} according as p is even or o...

It is an old problem in graph theory to test whether a graph contains a chordless cycle of length greater than three (hole) with a specific parity (even, odd). Studying the structure of graphs without odd holes has obvious implications for Berge’s strong perfect graph conjecture that states that a graph G is perfect if and only if neither G nor its complement contain an odd hole. Markossian, Ga...