A graph G is (m,k)-colourable if its vertices can be coloured with m colours such that the maximum degree of the subgraph induced on vertices receiving the same colour is at most k. The k-defective chromatic number χk(G) is the least positive integer m for which G is (m,k)colourable. Maddox proved that χk(G)+χk(Ḡ) ≤ 5 p 3k+4 whenever G is a triangle-free of order p and k ≥ 0 is an integer. Sima...

Frequency planning consists in allocating frequencies to the transmitters of a cellular network so as to ensure that no pair of transmitters interfere. We study the problem of reducing interference by modeling this by a radio k-labeling problem on graphs: For a graph G and an integer k ≥ 1, a radio k-labeling of G is an assignment f of non negative integers to the vertices of G such that |f(x) ...

In this paper, we introduce a new variation of list-colorings. For a graph G and for a given nonnegative integer t, a t-common list assignment of G is a mapping L which assigns each vertex v a set L(v) of colors such that given set of t colors belong to L(v) for every v ∈ V (G). The t-common list chromatic number of G denoted by cht(G) is defined as the minimum positive integer k such that ther...

The covering number c(K) of a convex body K is the least number of smaller homothetic copies of K needed to cover K . We provide new upper bounds for c(K) when K is centrally symmetric by introducing and studying the generalized α -blocking number βα 2 (K) of K . It is shown that when a centrally symmetric convex body K is sufficiently close to a centrally symmetric convex body K′ , then c(K) i...

In arrangements of pseudocircles (Jordan curves) the weight of a vertex (intersection point) is the number of pseudocircles that contain the vertex in its interior. We give improved upper bounds on the number of vertices of weight ≤ k in certain arrangements of pseudocircles in the plane.

In arrangements of pseudocircles (Jordan curves) the weight of a vertex (intersection point) is the number of pseudocircles that contain the vertex in its interior. We give improved upper bounds on the number of vertices of weight ≤ k in certain arrangements of pseudocircles in the plane. In particular, forbidding certain subarrangements we improve the known bound of 6n − 12 (cf. [2]) for verti...

It is known that any k-uniform family with covering number t has at most k t t-covers. In this paper, we give better upper bounds for the number of t-covers in k-uniform intersecting families with covering number t.

Let γ(k, p) denote Waring’s number (mod p) and δ(k, p) denote the ± Waring’s number (mod p). We use sum-product estimates for |nA| and |nA − nA|, following the method of Glibichuk and Konyagin, to estimate γ(k, p) and δ(k, p). In particular, we obtain explicit numerical constants in the Heilbronn upper bounds: γ(k, p) ≤ 83 k, δ(k, p) ≤ 20 k for any positive k not divisible by (p− 1)/2.

Let [Formula: see text] be a graph. A set [Formula: see text] is a distance k-dominating set of G if for every vertex [Formula: see text], [Formula: see text] for some vertex [Formula: see text], where k is a positive integer. The distance k-domination number [Formula: see text] of G is the minimum cardinality among all distance k-dominating sets of G. The first Zagreb index of G is defined as ...