In many applications of graph colouring the sizes of colour classes should not be too large. For example, in scheduling jobs (some of which could be performed at the same time), it is not good if the resulting schedule requires many jobs to occur at some specific time. An application of this type is discussed in [8]. A possible formalization of this restriction is the notion of equitable colour...

A proper vertex coloring of a graph G is equitable if the sizes of color classes differ by at most one. The equitable chromatic threshold χeq(G) of G is the smallest integer m such that G is equitably n-colorable for all n ≥ m. We show that for planar graphs G with minimum degree at least two, χeq(G) ≤ 4 if the girth of G is at least 10, and χeq(G) ≤ 3 if the girth of G is at least 14.

Because the partition coefficient is one of the most important parameters affecting microsegregation, the aim of this research is to experimentally analyse the partition coefficient in Al-Mg alloys. In order to experimentally measure the partition coefficient, a series of quenching experiments during solidification were carried out. For this purpose binary Al-Mg alloys containing 6.7 and 10.2 w...

Tasks partitioning is one of the most challenging issues in large scale parallel simulations. The simulation performance may vary dramatically if different partition schemes are used. This paper presents a spatial partition method named fine-grained spatial partition for parallel simulation of large scale wildfire. The fine-grained spatial partition is inspired by the four color theorem. It div...

For a positive integer k, a k-rainbow dominating function of a graph G is a function f from the vertex set V (G) to the set of all subsets of the set {1, 2, . . . , k} such that for any vertex v ∈ V (G) with f(v) = ∅ the condition ⋃ u∈N(v) f(u) = {1, 2, . . . , k} is fulfilled, where N(v) is the neighborhood of v. The 1-rainbow domination is the same as the ordinary domination. A set {f1, f2, ....

We study the diffusion of epidemics on networks that are partitioned into local communities. The gross structure of hierarchical networks of this kind can be described by a quotient graph. The rationale of this approach is that individuals infect those belonging to the same community with higher probability than individuals in other communities. In community models the nodal infection probabili...

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 ∑ x∈N[v] f (x) ≥ 1 for each v ∈ V (G), where N[v] is the closed neighborhood of v, then f is a signed dominating function on G. A set {f1, f2, . . . , fd} of signed dominating functions on Gwith the property that ∑d i=1 fi(x) ≤ 1 for each x ∈ V (G), is called a signed dominating fa...

Releasing the exact degree sequence of a graph for analysis may violate privacy. However, the degree sequence of a graph is an important summary statistic that is used in many statistical models. Hence a natural starting point is to release a private version of the degree sequence. A graphical degree partition is a monotonic degree sequence such that there exists a simple graph realizing the se...