Let G = (V,E) be a graph with n vertices and f : V (G) → {1, 2, . . . , n} be a bijective function. We define the minimum edge difference as ab(G, f) = min{|f(u)− f(v)| : (u, v) ∈ E}. We say that f is a k-antibandwidth labeling on G if ab(G, f) ≥ k. Let ab(G) = maxf{ab(G, f)}. We therefore investigate the lower bound of ab(G). In other words, to what extent can we maximize the minimum edge diff...

an equitable domination has interesting application in the contextof social networks. in a network, nodes with nearly equal capacitymay interact with each other in a better way. in the societypersons with nearly equal status, tend to be friendly. in thispaper, we introduce new variant of equitable domination of agraph. basic properties and some interesting results have beenobtained.

In this note, we derive an explicit formula for the equitable chromatic number of a complete n-partite graph Kp1 ;p2 ;:::;pn . Namely, if M is the largest integer such that pi (modM)¡ ⌈pi M ⌉ (i = 1; 2; : : : ; n) then e(Kp1 ;p2 ;:::;pn) = n ∑

We propose the following conjecture to generalize results of Pósa and Corrádi Hajnal. Let r, s be nonnegative integers and let G be a graph with |V (G)| ≥ 3r + 4s and minimal degree δ(G) ≥ 2r + 3s. Then G contains a collection of r + s vertex disjoint cycles, s of them with a chord. We prove the conjecture for r = 0, s = 2 and for s = 1. The corresponding extremal problem, to find the minimum n...