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Xiaodong Xu, Meilian Liang and Stanisław Radziszowski

Abstract

We present some new constructive upper bounds based on product graphs for generalized vertex Folkman numbers. They lead to new upper bounds for some special cases of generalized edge Folkman numbers, including the cases F e(K 3, K 4e; K 5) ≤ 27 and F e(K 4e, K 4e; K 5) ≤ 51. The latter bound follows from a construction of a K 5-free graph on 51 vertices, for which every edge coloring with two colors contains a monochromatic K 4e.

Open access

Xueliang Li and Xiaoyu Zhu

Abstract

For an r-regular graph G, we define an edge-coloring c with colors from {1, 2, . . . , k}, in such a way that any vertex of G is incident with at least one edge of each color. The multiset-color cm(v) of a vertex v is defined as the ordered tuple (a 1, a 2, . . . , ak), where ai (1 ≤ ik) denotes the number of edges of color i which are incident with v in G. Then this edge-coloring c is called a k-kaleidoscopic coloring of G if every two distinct vertices in G have different multiset-colors and in this way the graph G is defined as a k-kaleidoscope. In this paper, we determine the integer k for a complete graph Kn to be a k-kaleidoscope, and hence solve a conjecture in [P. Zhang, A Kaleidoscopic View of Graph Colorings, (Springer Briefs in Math., New York, 2016)] that for any integers n and k with nk + 3 ≥ 6, the complete graph Kn is a k-kaleidoscope. Then, we construct an r-regular 3-kaleidoscope of order (2r-1)-1 − 1 for each integer r ≥ 7, where r ≡ 3 (mod 4), which solves another conjecture in [P. Zhang, A Kaleidoscopic View of Graph Colorings, (Springer Briefs in Math., New York, 2016)] on the maximum order of r-regular 3-kaleidoscopes.

Open access

Stanislav Jendroľ and Lucia Kekeňáková

Abstract

A vertex coloring of a plane graph G is a facial rainbow coloring if any two vertices of G connected by a facial path have distinct colors. The facial rainbow number of a plane graph G, denoted by rb(G), is the minimum number of colors that are necessary in any facial rainbow coloring of G. Let L(G) denote the order of a longest facial path in G. In the present note we prove that rb(T)32L(T) for any tree T and rb(G)53L(G) for arbitrary simple graph G. The upper bound for trees is tight. For any simple 3-connected plane graph G we have rb(G) ≤ L(G) + 5.

Open access

Hui Jiang, Xueliang Li and Yingying Zhang

Abstract

A graph is said to be total-colored if all the edges and the vertices of the graph are colored. A total-coloring of a graph is a total monochromatically-connecting coloring (TMC-coloring, for short) if any two vertices of the graph are connected by a path whose edges and internal vertices have the same color. For a connected graph G, the total monochromatic connection number, denoted by tmc(G), is defined as the maximum number of colors used in a TMC-coloring of G. In this paper, we study two kinds of Erdős-Gallai-type problems for tmc(G) and completely solve them.

Open access

Magda Dettlaff, Joanna Raczek and Jerzy Topp

Abstract

The domination subdivision number sd(G) of a graph G is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number of G. It has been shown [10] that sd(T) ≤ 3 for any tree T. We prove that the decision problem of the domination subdivision number is NP-complete even for bipartite graphs. For this reason we define the domination multisubdivision number of a nonempty graph G as a minimum positive integer k such that there exists an edge which must be subdivided k times to increase the domination number of G. We show that msd(G) ≤ 3 for any graph G. The domination subdivision number and the domination multisubdivision number of a graph are incomparable in general, but we show that for trees these two parameters are equal. We also determine the domination multisubdivision number for some classes of graphs.

Open access

C. Balbuena, H. Galeana-Sánchez and M. Guevara

Abstract

Let D be a digraph of minimum in-degree at least 1. We prove that for any two natural numbers k, l such that 1 ≤ lk, the number of (k, l)-kernels of D is less than or equal to the number of (k, l)-kernels of any partial line digraph ℒD. Moreover, if l < k and the girth of D is at least l +1, then these two numbers are equal. We also prove that the number of semikernels of D is equal to the number of semikernels of ℒD. Furthermore, we introduce the concept of (k, l)-Grundy function as a generalization of the concept of Grundy function and we prove that the number of (k, l)-Grundy functions of D is equal to the number of (k, l)-Grundy functions of any partial line digraph ℒD.

Open access

Lei Su, Zongqiang Xie, Wenting Xu and Changming Zhao

Abstract

Mixed evergreen-deciduous broadleaved forest is the transitional type of evergreen broadleaved forest and deciduous broadleaved forest, and plays a unique eco-hydrologic role in terrestrial ecosystem. We investigated the spatio-temporal patterns of throughfall volume of the forest type in Shennongjia, central China. The results indicated that throughfall represented 84.8% of gross rainfall in the forest. The mean CV (coefficient of variation) of throughfall was 27.27%. Inter-event variability in stand-scale throughfall generation can be substantially altered due to changes in rainfall characteristics, throughfall CV decreased with increasing rainfall amount and intensity, and reached a quasi-constant level when rainfall amount reached 25 mm or rainfall intensity reached 2 mm h−1. During the leafed period, the spatial pattern of throughfall was highly temporal stable, which may result in spatial heterogeneity of soil moisture.

Open access

Judyta Bąk and Andrzej Kucharski

Abstract

We give a proposal of generalization of the Freese–Nation property for topological spaces. We introduce a few properties related to Freese–Nation property: FNS, FN, FNS*, FNI. This article presents some relationship between these concepts. We show that spaces with the FNS property satisfy ccc and any product of such spaces also satisfies ccc. We show that all metrizable spaces have the FN property.

Open access

Gian Luigi Forti

Abstract

Investigating Hyers–Ulam stability of the additive Cauchy equation with domain in a group G, in order to obtain an additive function approximating the given almost additive one we need some properties of G, starting from commutativity to others more sophisticated. The aim of this survey is to present these properties and compare, as far as possible, the classes of groups involved.

Open access

Memudu O. Olatinwo

Abstract

In this article, we establish some non-unique fixed point theorems of Ćirić’s type for (Φ, ψ)–hybrid contractive mappings by using a similar notion to that of the paper [M. Akram, A.A. Zafar and A.A. Siddiqui, A general class of contractions: A–contractions, Novi Sad J. Math. 38 (2008), no. 1, 25–33]. Our results generalize, extend and improve several ones in the literature.