Pancyclity when each cycle contains k chords

Pancyclicity when each cycle must pass exactly k Hamilton cycle chords

It is known that Θ(log n) chords must be added to an n-cycle to produce a pancyclic graph; for vertex pancyclicity, where every vertex belongs to a cycle of every length, Θ(n) chords are required. A possibly ‘intermediate’ variation is the following: given k, 1 ≤ k ≤ n, how many chords must be added to ensure that there exist cycles of every possible length each of which passes exactly k chords...

Optimal strategy of coupon subset collection when each package contains half of the coupons

The coupon subset collection is a generalization of the classical coupon collection, where instead of selecting (with replacement) a single coupon, a subset of at most k coupons (known as a “package”) is selected in each round. In this paper, we study how to design the collection of packages and assign probabilities to the packages, so as to minimize the expected number of rounds to collect all...

A 2-factor in which each cycle contains a vertex in a specified stable set

Let G be a graph with order n, and let k be an integer with 1 ≤ k ≤ n/3. In this article, we show that if σ2(G) ≥ n+ k− 1, then for any stable set S ⊆ V (G) with |S| = k, there exists a 2-factor with precisely k cycles C1, . . . , Ck such that |V (Ci) ∩ S| = 1 for all 1 ≤ i ≤ k and at most one of the cycles Ci has length strictly greater than three. The lower bound on σ2 is best possible.

Odd Graceful Labeling of Cycle with Parallel Pk Chords

A graph labeling is an assignment of integers to the vertices (or) edges or both subject to certain conditions. Graceful labeling was introduced by Rosa [6] while the concept of odd graceful labeling was introduced by Gnanajothi [2]. A Graph G (V, E), |V(G)| = p, |E(G)| = q is said to be odd graceful if there is an injection from V(G) to {0, 1, 2, .... 2q-1} such that when each edge xy is assig...

Gracefulness of a cycle with parallel Pk-chords