A given nonincreasing sequence D = (d1, d2, · · · , dn) is said to contain a (nonincreasing) repetition sequence D * = (di 1 , di 2 , · · · , di k) for some k ≤ n − 2 if all values of D − D * are distinct and for any di l ∈ D * there exists some dt ∈ D − D * such that di l = dt. For any pair of integers n and k with n ≥ k + 2, we investigate the existence… (More)
Consider the following two-person game on a graph G. Players I and II move alternatively to color a yet uncolored vertex of G properly using a pre-specified set of colors. Furthermore , Player II can only use the colors that have been used , unless he is forced to use a new color to guarantee that the graph is colored properly. The game ends when some… (More)
For any natural number k, a graph G is said to be pancyclic mod k if it contains a cycle of every length modulo k. In this paper, we show that every K 1,4-free graph G with minimum degree δ(G) ≥ k + 3 is pancyclic mod k and every claw-free graph G with δ(G) ≥ k + 1 is pancyclic mod k, which confirms Thomassen's conjecture  for claw-free graphs.
For two integers k(> 0) and s(≥ 0), a cycle of length l is called an (s mod k)-cycle if l ≡ s mod k. In this paper, the following conjecture of Chen, Dean, and Shreve  is proved: Every 2-connected graph with at least six vertices and minimum degree at least three contains a (2 mod 4)-cycle. 1 INTRODUCTION We use  for our notation and terminology. Our… (More)
In this paper, we investigate the sufficient conditions for a graph to contain a cycle (path) C such that G − V (C) is a disjoint union of cliques. In particular, sufficient conditions involving degree sum and neighborhood union are obtained.