A linear k-forest of an undirected graph G is a subgraph of G whose components are paths with lengths at most k. The linear k-arboricity Of G, denoted by lak(G), is the minimum number of linear k-forests needed to partition the edge set E(G) of G. In case that the lengths of paths are not restricted, we then have the linear arboricity ofG, denoted by la(G). In this paper, the exact values of th...

We prove the following theorems: 1. If X has strong measure zero and if Y has strong first category, then their algebraic sum has property s0. 2. If X has Hurewicz’s covering property, then it has strong measure zero if, and only if, its algebraic sum with any first category set is a first category set. 3. If X has strong measure zero and Hurewicz’s covering property then its algebraic sum with...

The local chromatic number of a graph was introduced in [13]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the stable Kneser (or Schrijver) graphs; Mycielski graphs, and their generalizations; and...

Wind speed modeling and prediction plays a critical role in wind related engineering studies. However, since the data have random behavior, it is difficult to apply statistical approaches with apriori and deterministic parameters. On the other hand, wind speed data have an important feature; extreme transitions from a wind state to a far different one are rare. Therefore, behavioral modeling is...

The Narumi-Katayama index of a graph G is equal to the product of degrees of all the vertices of G. In this paper, we examine the NarumiKatayama index of some derived graphs such as a Mycielski graph, subdivision graphs, double graph, extended double cover graph, thorn graph, subdivision vertex join and edge join graphs.

In an earlier paper, the present authors (2013) [1] introduced the alternating chromatic number for hypergraphs and used Tucker’s Lemma, an equivalent combinatorial version of the Borsuk-Ulam Theorem, to show that the alternating chromatic number is a lower bound for the chromatic number. In this paper, we determine the chromatic number of some families of graphs by specifying their alternating...

For a graph G and its spanning tree T the backbone chromatic number, BBC(G,T ), is defined as the minimum k such that there exists a coloring c : V (G) → {1, 2, . . . , k} satisfying |c(u) − c(v)| ≥ 1 if uv ∈ E(G) and |c(u)− c(v)| ≥ 2 if uv ∈ E(T ). Broersma et al. [1] asked whether there exists a constant c such that for every triangle-free graphG with an arbitrary spanning tree T the inequali...

Let X be a Polish space, that is, a separable, completely metrizable topological space. We always assume that X is non-empty and perfect, that is, X has no isolated points. An old theorem of Mycielski (see [8]) says that given a sequence of meager sets …Bn Í X k…n†: n < q†, where 1 < k…n† < q, there exists a perfect set P Í X such that P is free for each Bn , that is, P k…n† Ç Bn Í f…x0 ; . . ....

The local chromatic number of a graph was introduced in [12]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and...

Let n(G) denote the number of vertices of a graph G and let (G) be the independence number of G, the maximum number of pairwise nonadjacent vertices of G. The Hall ratio of a graph G is defined by (G)=max { n(H) (H) : H ⊆ G } , where the maximum is taken over all induced subgraphs H of G. It is obvious that every graph G satisfies (G) (G) (G) where and denote the clique number and the chromatic...