Let D be a finite and simple digraph with vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑ x∈N−[v] f(x) ≥ k for each v ∈ V (D), where N−[v] consists of v and all vertices of D from which arcs go into v, then f is a signed k-dominating function on D. A set {f1, f2, . . . , fd} of distinct signed k-dominating functions of D with the property tha...

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...

An (I,Fd)-partition of a graph is a partition of the vertices of the graph into two sets I and F , such that I is an independent set and F induces a forest of maximum degree at most d. We show that for all M < 3 and d ≥ 2 3−M − 2, if a graph has maximum average degree less than M , then it has an (I,Fd)-partition. Additionally, we prove that for all 8 3 ≤ M < 3 and d ≥ 1 3−M , if a graph has ma...

We propose an approach to achieving different K-partition for preserving privacy by using the multiconstraint anonymous parameter design method based on the attribute significance of rough set, in order to reduce the imbalance phenomenon between the privacy protection and data availability caused by adopting the same anonymous intensity. In this approach, taking into account the significance of...

Let G be a finite and simple graph with vertex set V (G), and let f: V (G)→ {−1, 1} be a two-valued function. If k > 1 is an integer and ∑ x∈N[v] f(x) > k for each v ∈ V (G), where N [v] is the closed neighborhood of v, then f is a signed k-dominating function on G. A set {f1, f2, . . . , fd} of signed kdominating functions on G with the property that ∑ d i=1 fi(x) 6 k for each x ∈ V (G), is ca...

This paper copes with the issue of adaptive partitioning in the context of Iterated Functions Systems (IFS) image coding. The main concern is to provide some insights on the optimum degree of partition adaptivity. The key point is to point out the proper balance of information between the partition representation and the transformation parameters of the system. Several systems involving diieren...