Further results on the reciprocal degree distance of graphs

The reciprocal degree distance of a simple connected graph \(G=(V_G, E_G)\) is defined as \(\bar{R}(G)=\sum _{u,v \in V_G}(\delta _G(u)+\delta _G(v))\frac{1}{d_G(u,v)}\), where \(\delta _G(u)\) is the vertex degree of \(u\), and \(d_G(u,v)\) is the distance between \(u\) and \(v\) in \(G\). The reciprocal degree distance is an additive weight version of the Harary index, which is defined as \(H(G)=\sum _{u,v \in V_G}\frac{1}{d_G(u,v)}\). In this paper, the extremal \(\bar{R}\)-values on several types of important graphs are considered. The graph with the maximum \(\bar{R}\)-value among all the simple connected graphs of diameter \(d\) is determined. Among the connected bipartite graphs of order \(n\), the graph with a given matching number (resp. vertex connectivity) having the maximum \(\bar{R}\)-value is characterized. Finally, sharp upper bounds on \(\bar{R}\)-value among all simple connected outerplanar (resp. planar) graphs are determined.     

Sparse multipartite graphs as partition universal for graphs with bounded degree

For graphs G and H, let \(G\rightarrow (H,H)\) signify that any red/blue edge coloring of G contains a monochromatic H as a subgraph. Denote \(\mathcal {H}(\Delta ,n)=\{H:|V(H)|=n,\Delta (H)\le \Delta \}\). For any \(\Delta \) and n, we say that G is partition universal for \(\mathcal {H}(\Delta ,n)\) if \(G\rightarrow (H,H)\) for every \(H\in \mathcal {H}(\Delta ,n)\). Let \(G_r(N,p)\) be the random spanning subgraph of the complete r-partite graph \(K_r(N)\) with N vertices in each part, in which each edge of \(K_r(N)\) appears with probability p independently and randomly. We prove that for fixed \(\Delta \ge 2\) there exist constants r, B and C depending only on \(\Delta \) such that if \(N\ge Bn\) and \(p=C(\log N/N)^{1/\Delta }\), then asymptotically almost surely \(G_r(N,p)\) is partition universal for \(\mathcal {H}(\Delta ,n)\).     

An efficient algorithm for distance total domination in block graphs

The \(k\)-distance total domination problem is to find a minimum vertex set \(D\) of a graph such that every vertex of the graph is within distance \(k\) from some vertex of \(D\) other than itself, where \(k\) is a fixed positive integer. In the present paper, by using a labeling method, we design an efficient algorithm for solving the \(k\)-distance total domination problem on block graphs, a superclass of trees.     

Distance domination in graphs with given minimum and maximum degree

For an integer \(k \ge 1\), a distance k-dominating set of a connected graph G is a set S of vertices of G such that every vertex of V(G) is at distance at most k from some vertex of S. The distance k-domination number \(\gamma _k(G)\) of G is the minimum cardinality of a distance k-dominating set of G. In this paper, we establish an upper bound on the distance k-domination number of a graph in terms of its order, minimum degree and maximum degree. We prove that for \(k \ge 2\), if G is a connected graph with minimum degree \(\delta \ge 2\) and maximum degree \(\Delta \) and of order \(n \ge \Delta + k - 1\), then \(\gamma _k(G) \le \frac{n + \delta - \Delta }{\delta + k - 1}\). This result improves existing known results.     

Adjacent vertex distinguishing edge-colorings of graphs with smaller maximum average degree

An adjacent vertex distinguishing edge-coloring of a graph G is a proper edge coloring of G such that any pair of adjacent vertices are incident to distinct sets of colors. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring of G is denoted by χ′ _a (G). Let mad(G)\mathop{\mathrm{mad}}(G) and Δ denote the maximum average degree and the maximum degree of a graph G, respectively.     

Minimum statuses of connected graphs with fixed maximum degree and order

The status of a vertex in a connected graph is the sum of distances between the vertex and all vertices. The minimum status of a connected graph is the minimum of statuses of all vertices of this graph. In this paper we obtain the sharp lower bound and the sharp upper bound on the minimum status of a connected graph with maximum degree k and order n. All the graphs attaining the lower bound are obtained, and a necessary condition is given for those graphs attaining the upper bound.     

The minimum value of geometric-arithmetic index of graphs with minimum degree 2

The geometric-arithmetic index was introduced in the chemical graph theory and it has shown to be applicable. The aim of this paper is to obtain the extremal graphs with respect to the geometric-arithmetic index among all graphs with minimum degree 2. Let G(2, n) be the set of connected simple graphs on n vertices with minimum degree 2. We use linear programming formulation and prove that the minimum value of the first geometric-arithmetic \((GA_{1})\) index of G(2, n) is obtained by the following formula:

$$\begin{aligned} GA_1^* = \left\{ \begin{array}{ll} n&{}\quad n \le 24, \\ \mathrm{{24}}\mathrm{{.79}}&{}\quad n = 25, \\ \frac{{4\left( {n - 2} \right) \sqrt{2\left( {n - 2} \right) } }}{n}&{}\quad n \ge 26. \\ \end{array} \right. \end{aligned}$$

     

Partial degree bounded edge pacing problem for graphs and $$$$-uniform hypergraphs

Given a graph \(G=(V,E)\) and a non-negative integer \(c_u\) for each \(u\in V\), partial degree bounded edge packing problem is to find a subgraph \(G^{\prime }=(V,E^{\prime })\) with maximum \(|E^{\prime }|\) such that for each edge \((u,v)\in E^{\prime }\), either \(deg_{G^{\prime }}(u)\le c_u\) or \(deg_{G^{\prime }}(v)\le c_v\). The problem has been shown to be NP-hard even for uniform degree constraint (i.e., all \(c_u\) being equal). In this work we study the general degree constraint case (arbitrary degree constraint for each vertex) and present two combinatorial approximation algorithms with approximation factors \(4\) and \(2\). Then we give a \(\log _2 n\) approximation algorithm for edge-weighted version of the problem and an efficient exact algorithm for edge-weighted trees with time complexity \(O(n\log n)\). We also consider a generalization of this problem to \(k\)-uniform hypergraphs and present a constant factor approximation algorithm based on linear programming using Lagrangian relaxation.     

List edge and list total coloring of planar graphs with maximum degree 8

Let \(G\) be a planar graph with maximum degree \(\varDelta \ge 8\) and without chordal 5-cycles. Then \(\chi '_{l}(G)=\varDelta \) and \(\chi ''_{l}(G)=\varDelta +1\).     

Chromatic number of distance graphs generated by the sets {2,3,x,y}

Let D be a set of positive integers. The distance graph generated by D has all integers ? as the vertex set; two vertices are adjacent whenever their absolute difference falls in D. We completely determine the chromatic number for the distance graphs generated by the sets D={2,3,x,y} for all values x and y. The methods we use include the density of sequences with missing differences and the parameter involved in the so called “lonely runner conjecture”. Previous results on this problem include: For x and y being prime numbers, this problem was completely solved by Voigt and Walther (Discrete Appl. Math. 51:197–209, 1994); and other results for special integers of x and y were obtained by Kemnitz and Kolberg (Discrete Math. 191:113–123, 1998) and by Voigt and Walther (Discrete Math. 97:395–397, 1991).     

Total coloring of 1-toroidal graphs with maximum degree at least 11 and no adjacent triangles

A total coloring of a graph G is an assignment of colors to the vertices and the edges of G such that every pair of adjacent/incident elements receive distinct colors. The total chromatic number of a graph G, denoted by \(\chi ''(G)\), is the minimum number of colors in a total coloring of G. The well-known total coloring conjecture (TCC) says that every graph with maximum degree \(\Delta \) admits a total coloring with at most \(\Delta + 2\) colors. A graph is 1-toroidal if it can be drawn in torus such that every edge crosses at most one other edge. In this paper, we investigate the total coloring of 1-toroidal graphs, and prove that the TCC holds for the 1-toroidal graphs with maximum degree at least 11 and some restrictions on the triangles. Consequently, if G is a 1-toroidal graph with maximum degree \(\Delta \) at least 11 and without adjacent triangles, then G admits a total coloring with at most \(\Delta + 2\) colors.     

The thickness of the complete multipartite graphs and the join of graphs

The thickness of a graph is the minimum number of planar spanning subgraphs into which the graph can be decomposed. It is known for relatively few classes of graphs, compared to other topological invariants, e.g., genus and crossing number. For the complete bipartite graphs, Beineke et al. (Proc Camb Philos Soc 60:1–5, 1964) gave the answer for most graphs in this family in 1964. In this paper, we derive formulas and bounds for the thickness of some complete k-partite graphs. And some properties for the thickness for the join of two graphs are also obtained.     

Visual cryptography on graphs

In this paper, we consider a new visual cryptography scheme that allows for sharing of multiple secret images on graphs: we are given an arbitrary graph (V,E) where every node and every edge are assigned an arbitrary image. Images on the vertices are “public” and images on the edges are “secret”. The problem that we are considering is how to make a construction such that when the encoded images of two adjacent vertices are printed on transparencies and overlapped, the secret image corresponding to the edge is revealed. We define the most stringent security guarantees for this problem (perfect secrecy) and show a general construction for all graphs where the cost (in terms of pixel expansion and contrast of the images) is proportional to the chromatic number of the cube of the underlying graph. For the case of bounded degree graphs, this gives us constant-factor pixel expansion and contrast. This compares favorably to previous works, where pixel expansion and contrast are proportional to the number of images.     

Further thoughts on the minimax principle

A two-person non-zero sum game may have two distinct solutions: the maximin strategy and the equilibrium point. The choice of solution depends on the players' objectives and it can be shown that, although strategies may be found that are better than the maximin solution in terms of payoff, restraint and co-operation are needed for players to achieve such benefits.     

Profile minimization on compositions of graphs

The profile minimization problem arose from the study of sparse matrix technique. In terms of graphs, the problem is to determine the profile of a graph G which is defined as $$P(G)=\min\limits_{f}\sum\limits_{v\in V(G)}\max\limits_{x\in N[v]}(f(v)-f(x)),$$ where f runs over all bijections from V(G) to {1,2,…,|V(G)|} and N[v]={v}∪{x∈V(G):xv∈E(G)}. This is equivalent to the interval graph completion problem, which is to find a super-graph of a graph G with as few number of edges as possible. The purpose of this paper is to study the profiles of compositions of two graphs.     

Fast searching games on graphs

Given a graph, suppose that intruders hide on vertices or along edges of the graph. The fast searching problem is to find the minimum number of searchers required to capture all the intruders satisfying the constraint that every edge is traversed exactly once and searchers are not allowed to jump. In this paper, we prove lower bounds on the fast search number. We present a linear time algorithm to compute the fast search number of Halin graphs and their extensions. We present a quadratic time algorithm to compute the fast search number of cubic graphs.     

Further Thoughts on the Utility of Risk Matrices

Risk matrices are commonly encountered devices for rating hazards in numerous areas of risk management. Part of their popularity is predicated on their apparent simplicity and transparency. Recent research, however, has identified serious mathematical defects and inconsistencies. This article further examines the reliability and utility of risk matrices for ranking hazards, specifically in the context of public leisure activities including travel. We find that (1) different risk assessors may assign vastly different ratings to the same hazard, (2) even following lengthy reflection and learning scatter remains high, and (3) the underlying drivers of disparate ratings relate to fundamentally different worldviews, beliefs, and a panoply of psychosocial factors that are seldom explicitly acknowledged. It appears that risk matrices when used in this context may be creating no more than an artificial and even untrustworthy picture of the relative importance of hazards, which may be of little or no benefit to those trying to manage risk effectively and rationally.