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Coloring the vertices of a particular graph has often been motivated by its utility to various applied fields and its mathematical interest. A dynamic coloring of a graph G is a proper coloring of the vertex set V(G) such that for each vertex of degree at least 2, its neighbors receive at least two distinct colors. A dynamic k-coloring of a graph is a dynamic coloring with k colors. A dynamic k-coloring is also called a conditional (k, 2)-coloring. The smallest integer k such that G has a dynamic k-coloring is called the dynamic chromatic number χ d (G) of G. In this paper, we investigate the dynamic chromatic number for the line graph of sunlet graph and middle graph, total graph and central graph of sunlet graphs, paths and cycles. Also, we find the dynamic chromatic number for Mycielskian of paths and cycles and the join graph of paths and cycles. [ABSTRACT FROM AUTHOR]

We study a version of online edge coloring, where the goal is to color as many edges as possible using only a given number, k, of available colors. All of our results are with regard to competitive analysis. Previous attempts to identify optimal algorithms for this problem have failed, even for bipartite graphs. Thus, in this paper, we analyze even more restricted graph classes, paths and trees. For paths, we consider k=2, and for trees, we consider any k≥2. We prove that a natural greedy algorithm called FIRST-FIT is optimal among deterministic algorithms, on paths as well as trees. For paths, we give a randomized algorithm, which is optimal and better than the best possible deterministic algorithm. For trees, we prove that to obtain a better competitive ratio than FIRST-FIT, the algorithm would have to be both randomized and unfair (i.e., reject edges that could have been colored), and even such algorithms cannot be much better than FIRST-FIT. [ABSTRACT FROM AUTHOR]

Let D be a digraph on $\{v_{1},\ldots, v_{n}\}$ . Then the sequence $\{ (d^{+}(v_{1}), d^{-}(v_{1})), \ldots, (d^{+}(v_{n}), d^{-}(v_{n}))\}$ is called the degree sequence of D. For any given sequence of pairs of integers $\mathbf{d}=\{(d_{1}^{+}, d_{1}^{-}), \ldots, (d_{n}^{+}, d_{n}^{-})\}$ , if there exists a k-arc strongly connected digraph D such that d is the degree sequence of D, then d is realizable and D is a realization of d. In this paper, characterizations for k-arc-connected realizable sequences and realizable sequences with arc-connectivity exactly k are given. [ABSTRACT FROM AUTHOR]

The possibility of diagnosis of the multiprocessor systems under multiple, up to n−2, failed processors within the framework of the Preparata-Metze-Chien model and method was analyzed. The complete system diagnosis, that is, determination of the good-failed state of all processors, is sometimes possible using the proposed method of analysis of the system diagnostic graph. More precisely, the conditions were determined for impossibility of system diagnosis. [ABSTRACT FROM AUTHOR]

In this paper, we determine the unique graph whose least signless Laplacian eigenvalue attains the minimum among all non-bipartite unicyclic graphs of order n with maximum degree Δ and among all non-bipartite connected graphs of order n with maximum degree Δ, respectively. [ABSTRACT FROM AUTHOR]

The sets with continuous selection from near-best approximations and the monotone path-connected sets are studied; several examples of such sets are also considered. [ABSTRACT FROM AUTHOR]

The problem of two edge-disjoint paths is to identify two paths $$Q_1$$ and $$Q_2$$ from source $$s \in V$$ to target $$t \in V$$ without any common arc in a directed connected graph $$G=(V, E)$$ . In this paper, we present an adaptive stabilizing algorithm for finding a pair of edge-disjoint paths from s to t in G in O( D) rounds with state-space complexity of $$O(log\; n)$$ bits per process, where n is the number of nodes and D is the diameter of the graph. The proposed algorithm is optimal with respect to its time complexity, and the total length of the shortest paths. In addition, it can also be used to solve the problem for undirected graphs. Since the proposed algorithm is stabilizing, it does not require initialization and is capable of withstanding transient faults. We view a fault that perturbs the state of the system but not its program as a transient fault. In addition, the proposed algorithm is adaptive since it is capable of dealing with topology changes in the form of addition/removal of arcs and/or nodes as well as changes in the directions of arcs provided that two edge-disjoint paths between s and t exist after the topology change. [ABSTRACT FROM AUTHOR]

We consider in this paper the problems of finding the elementary shortest and longest paths on a graph containing negative and positive cycles. These problems are NP-hard. We propose exact algorithms based on mixed integer programming for their solution, employing different valid inequalities. Moreover, we propose decomposition techniques which are very efficient for cases with special structure. Experimental results show the efficiency of our algorithms compared with state of the art exact algorithms. [ABSTRACT FROM AUTHOR]

A hop dominating set of a graph G is a set D of vertices of G if for every vertex of V (G) \ D, there exists u ∈ D such that d(u, v) = 2. The hop domination number of a graph G, denoted by γh(G), is the minimum cardinality of a hop dominating set of G. We prove that for every tree T of order n with l leaves and s support vertices we have (n − l − s + 4)/3 ≤ γh(G) ≤ n/2, and characterize the trees attaining each of the bounds. [ABSTRACT FROM AUTHOR]

This paper proposes an approach to parallel filling path generation based on adaptive gaps for fused deposition modeling (FDM) with full consideration of technology characteristics of this promising fabricating method. At first, analysis on the similarities and differences between FDM and the conventional milling manufacturing is conducted from the tool-path planning perspective. Meanwhile, some specific features of FDM are analyzed in detail to point out three critical issues in the path planning process which is correlated with the fabricating quality and modeling accuracy. These issues are described specifically by means of theoretical analysis and practical fabrication: large number of uncut paths, sharp corners in the filling path, and overfill and underfill phenomena which are out of the constant gap between parallel paths. The proposed tool-path generation method determines the inclination of the reference line based on less number of sharp corners in the paths firstly. Then, the filling area is divided with three stages. The pre-sweeping stage employs a group of paralleling lines with pre-determined gaps and optimized inclination to intersect the filling boundaries and stores the intersections between them, the second stage adjusts some sweeping line segments which satisfy several specific conditions, and the last stage divides the filling areas into several sub-regions according to the selected sweeping lines. Subsequently, the sweeping lines within each sub-region are adjusted adaptively to uniform gaps individually. At last, some adjacent sub-paths that can be connected to decrease the number of uncut paths are linked. Relative performance evaluation and experimental analyses are provided to demonstrate that the proposed methods can enhance the fabricating quality significantly with some improvements in the machining efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we present trapping issues of weight-dependent walks on weighted hierarchical networks which are based on the classic scale-free hierarchical networks. Assuming that edge's weight is used as local information by a random walker, we introduce a biased walk. The biased walk is that a walker, at each step, chooses one of its neighbours with a probability proportional to the weight of the edge. We focus on a particular case with the immobile trap positioned at the hub node which has the largest degree in the weighted hierarchical networks. Using a method based on generating functions, we determine explicitly the mean first-passage time (MFPT) for the trapping issue. Let parameter a (0 < a < 1) be the weight factor. We show that the efficiency of the trapping process depends on the parameter a; the smaller the value of a, the more efficient is the trapping process. [ABSTRACT FROM AUTHOR]

To solve minimum exposure path (MEP) problem in wireless sensor networks more efficiently, this work proposes an algorithm called target guiding self-avoiding random walk with intersection (TGSARWI), which mimics the behavior of a group of random walkers that seek path to their destinations in a strange area. Target guiding leads random walkers move toward their end points, while self-avoiding prevents them from taking roundabout routes. Route intersections further accelerate the speed of seeking connected paths. Dijkstra algorithm (DA) is applied to solve MEP problem in a sub-network formed by multiple connected paths that walkers generate (called TGSARWI DA). Simulations show that the path exposure found by TGSARWI DA is very close to that by DA in the global network (Global DA), whereas the time complexity of computation is much lower. Compared with existing heuristic algorithms such as physarum optimization algorithm (POA), our algorithm shows higher generality and efficiency. This algorithm also exhibits good robustness to the fluctuations of parameters. Our algorithm could be very useful for the solution to MEP problem in fields with large- or high-density sensors. [ABSTRACT FROM AUTHOR]