36089150
Hamiltonicity and circular distance two labellings
Abstract A k -circular distance two labelling (or k -c-labelling) of a graph G is a vertex-labelling such that the circular difference ( mod k ) of the labels is at least two for adjacent vertices, and at least one for vertices at distance two. Given G , denote σ(G) the minimum k for which there exists a k -c-labelling of G . Suppose G has n vertices, we prove σ(G)⩽n if G c is Hamiltonian; and σ(G)=n+p v (G c ) otherwise, where p v (G) is the path covering number of G . We give exact values of σ(G) for some families of graphs such that G c is Hamiltonian, and discuss injective k -c-labellings especially for joins and unions of graphs.
1708763
Multilevel Distance Labelings for Paths and Cycles
For a graph $G$, let $\diam(G)$ denote the diameter of $G$. For any two vertices $u$ and $v$ in $G$, let $d(u, v)$ denote the distance between $u$ and $v$. A multilevel distance labeling (or distance labeling) for $G$ is a function $f$ that assigns to each vertex of $G$ a nonnegative integer such that for any vertices $u$ and $v$, $|f(u)-f(v)| \geq \diam(G) - d_G(u, v) +1$. The span of $f$ is the largest number in $f(V)$. The radio number of $G$, denoted by $rn(G)$, is the minimum span of a distance labeling for $G$. In this paper, we completely determine the radio numbers for paths and cycles.
30709401
T-graphs and the channel assignment problem
Abstract T -colorings arose from the channel assignment problem in communications. Given a finite set T of non-negative integers, a T -coloring of a simple graph G is an assignment of a non-negative integer (channel) on every vertex of G , such that the difference of channels of two adjacent vertices does not fall in T . The T -span of G , denoted by sp T ( G ), is the minimum span among all possible T -colorings of G , where the span of a T -coloring is the difference between the largest and smallest channels used. This article applies T -graphs to explore the sets T that belong to these two collections: G = { T : greedy (or first-fit) algorithm provides T -spans for all complete graphs}, and E = { T : spt ( G ) = sp T ( K χ ( G ) ) for all graphs G , where χ is the chromatic number}. Given T , its T -graph has the vertex set of all non-negative integers so that two vertices are adjacent if their difference does not fall in T . We show that for any a ∈ Z + , the T -graph of aT is isomorphic to a disjoint product of a copies of the T -graph of T , where aT is the set obtained by multiplying every element of T by a . Based on this characterization, the following two results are attained. For any a ∈ Z + , T ∈ E if and only if aT ∈ E , and T ∈ G if and only if aT ∈ G . The second fact has been proven by Cozzens and Roberts (1991) from a different approach. We will completely solve the family of sets T = {0, s , s + 1,…, l } by providing a different proof of the fact T ∈ G (Tesman, 1989), and showing that T ∈ E if and only if l is a multiple of s . In addition, complete solutions for a more general family are obtained: for any a , s , l ∈ Z + with s ⩽ l , T = {0, as , a ( s + 1), a ( s + 2),…, al } ∪ A ∈ E , and T ϵ E if and only if l = ms for some m , where A ⊆{ as + 1, as + 2, …, al - 1}.
16699960
Circular chromatic number for iterated Mycielski graphs
Abstract For a graph G , let M( G ) denote the Mycielski graph of G . The t th iterated Mycielski graph of G , M t ( G ), is defined recursively by M 0 ( G )= G , and M t ( G )=M(M t −1 ( G )) for t ⩾1. Let χ c ( G ) denote the circular chromatic number of G . We prove two main results: (1) If G has a universal vertex x , then χ c (M( G ))= χ (M( G )) if χ c (G−x)>χ(G)− 1 2 and G is not a star, otherwise χ c ( M (G))=χ( M (G))− 1 2 ; and (2) χ c (M t ( K m ))= χ (M t ( K m )) if m ⩾2 t −1 +2 t −2 and t ⩾2.
38675279
On-Line Learning Control for Discrete Nonlinear Systems Via an Improved ADDHP Method
This paper mainly discusses a generic scheme for on-line adaptive critic design for nonlinear system based on neural dynamic programming (NDP), more exactly, an improved action-depended dual heuristic dynamic programming (ADDHP) method. The principal merit of the proposed method is to avoid the model neural network which predicts the state of next time step, and only use current and previous states in the method, as makes the algorithm more suitable for real-time or on-line application for process control. In this paper, convergence proof of the method will also be given to guarantee the control to reach the optimal. At last, simulation result verifies the performance.
6282895
Timing analysis including clock skew
Clock skew is an increasing concern for high-speed circuit designers. Circuit designers use transparent latches and skew-tolerant domino circuits to hide clock skew from the critical path and take advantage of shared portions of the clock network to budget less skew between nearby elements than across the entire die, but current timing analysis algorithms do not handle correlated clock skews. This paper extends the Sakallah-Mudge-Olukotun (SMO) latch-based timing analysis to include different amounts of clock skew between different elements. The key change is that departure times from each latch must be defined with respect to launching clocks so that the skew between the launching and receiving clocks can be determined at each receiver. The exact analysis leads to an explosion in the number of timing constraints, but most constraints are not tight in practical situations and a modified version of the Szymanski-Shenoy relaxation algorithm gives exact results with only a small increase in runtime. The timing analysis formulation also captures the effects of skew on edge-triggered flip-flops, domino circuits, and min-delay constraints. Our exact algorithm, applied to a supercomputer node controller with over 12000 clocked elements, finds the system can run 50-90 ps faster than a single skew analysis would predict and requires searching fewer than 4% more latch departures than conventional algorithms. With the less conservative skew budgets enabled by better timing analysis, we expect clocked systems will remain viable to multi-GHz frequencies.
14407932
Automatic measurement of quality metrics for colonoscopy videos
Colonoscopy is the accepted screening method for detection of colorectal cancer or its precursor lesions, colorectal polyps. Indeed, colonoscopy has contributed to a decline in the number of colorectal cancer related deaths. However, not all cancers or large polyps are detected at the time of colonoscopy, and methods to investigate why this occurs are needed. We present a new computer-based method that allows automated measurement of a number of metrics that likely reflect the quality of the colonoscopic procedure. The method is based on analysis of a digitized video file created during colonoscopy, and produces information regarding insertion time, withdrawal time, images at the time of maximal intubation, the time and ratio of clear versus blurred or non-informative images, and a first estimate of effort performed by the endoscopist. As these metrics can be obtained automatically, our method allows future quality control in the day-to-day medical practice setting on a large scale. In addition, our method can be adapted to other healthcare procedures. Last but not least, our method may be useful to assess progress during colonoscopy training, or as part of endoscopic skills assessment evaluations.
17515399
Sizes of Graphs with Fixed Orders and Spans for Circular-Distance-Two Labelings
A k-circular-distance-two labeling (or k-c-labeling) of a simple graph G is a vertex-labeling, using the labels 0, 1, 2, · · · , k − 1, such that the “circular difference” (mod k) of the labels for adjacent vertices is at least two, and for vertices of distance-two apart is at least one. The σ-number, σ(G), of a graph G is the minimum k of a k-c-labeling of G. For any given positive integers n and k, let G(n, k) denote the set of graphs G on n vertices and σ(G) = k. We determine the maximum size (number of edges) and the minimum size of a graph G ∈ G(n, k). Furthermore, we prove that for any value p between the maximum and the minimum size, there exists a graph G ∈ G(n, k) of size p. These results are analogues of the ones by Georges and Mauro [4] on distance-two labelings.