Maximum Degree

We give a randomized algorithm that properly colors the vertices of a triangle-free graph G on n vertices using O(\Delta(G)/ log \Delta(G)) colors, where \Delta(G) is the maximum degree of G. The algorithm takes O(n\Delta2(G)log\Delta(G)) time and succeeds with high probability, provided \Delta(G) is greater than log^{1+{\epsilon}}n for a positive constant {\epsilon}. The number of colors is best possible up to a constant factor for triangle-free graphs. As a result this gives an algorithmic proof for a sharp upper bound of the chromatic number of a triangle-free graph, the existence of which was previously established by Kim and Johansson respectively.

The domatic number of a graph G is the maximum number of dominating sets into which the vertex set of G can be partitioned. We show that the domatic number of a random r-regular graph is almost surely at most r, and that for 3-regular random graphs, the domatic number is almost surely equal to 3. We also give a lower bound on the domatic number of a graph in terms of order, minimum degree and maximum degree. As a corol- lary, we obtain the result that the domatic number of an r-regular graph is at least (r + 1)/(3ln(r + 1)).

Abstract: This research studied the effects of torrefaction temperature (250-250 C) and time (30-120 minutes) on elemental composition and energy properties changes in corn stover. Torrefied material was analyzed for moisture content, moisture-free carbon (%), hydrogen ...

A Hamiltonian cycle in a graph G is a simple cycle in which each vertex of G appears ex- actly once. The Hamiltonian cycle problem involves testing whether a Hamiltonian cycle exists in a graph, and finds one if such a cycle does exist. It is well known that the Hamiltonian cycle problem is one of the classic NP-complete problems on general graphs. Shih et al. solved the Hamiltonian cycle problem on circular-arc graphs in O(n2 log n) time (36), where n is the number of vertices of the input graph. Whether there exists a more efficient algorithm for solving the Hamiltonian cycle problem on circular-arc graphs has been opened for a decade. In this paper, we present an O(∆n)-time algorithm to solve it, where ∆ denotes the maximum degree of the input graph. All graphs considered in this paper are finite and undi- rected, without loops or multiple edges. Let G =( V, E) be a graph with vertex set V and edge set E. Through- out this paper, let m, n, and ∆ denote the number of edges, the nu...

Abstract. We present approximation algorithms for the unsplittable flow problem (UFP) in undirected graphs. As is standard in this line of research, we assume that the maximum demand is at most the minimum capacity. We focus on the non-uniform capacity case in which the edge ...

A graph G k-chordal, if it does not contain chordless cycles of length larger than k. The chordality cl of a graph G is the minimum k for which G is k-chordal. The degeneracy or the width of a graph is the maximum min-degree of any of its subgraphs. Our results are the following: 1. The problem of treewidth remains NP-complete when restricted on graphs with small maximum degree. 2. An upper bound is given for the treewidth of a graph as a function of its maximum degree and chordality. A consequence of this result is that when maximum degree and chordality are xed constants, then there is a linear algorithm for treewidth and a polynomial algorithm for pathwidth. 3. For any constant s 1, it is shown that any (s + 2)-chordal graph with degeneracy d contains a 1 2 -separator of size O((dn) s 1 s ), computable in linear time. Our results extent the many applications of the separator theorems in [1, 33, 34] to the class of k-chordal graphs. Several natural classes of graphs have small cho...

A statistical comparison has been made between gravity field parameters derived from different global geopotential models (GGMs) and observed gravity anomalies, gravimetric geoid and GPS-Leveling data. The motivation behind this study is the selection of best possible global geopotential model that best matches statistically with the local observed data in Pakistan. This will facilitate in decreasing the load on observed data for the development of regional gravimetric geoid in remove-compute-restore technique when used in the Stokes's integral for computation of the residual part. It is observed that combined geopotential models such as EGM96 and PGM200A, EIGEN-GL04C and EIGEN-CG03C reflect the better match in the total spectral range of gravity and GPS-Leveling data. Results of the precise local geoid model also indicate similar characteristics. A very-high-degree model "EGM2008" (degree/order 2160) exhibits relatively superior statistical fit with observed ground data in Pakistan region. For satellite-only models an increasing trend in the standard deviation can be seen with maximum of about ˜4 m in difference between GPS-Leveling and corresponding GGM's geoid with increase in the order from 50 to 120 and then it decreases afterwards. However, for the EIGEN-CHAMP03SP, standard deviation saturates to a value of 3.4 m. This is an indication of contamination in the long to medium wavelength part, i.e. 50-100° for the satellite-only models. Moreover, the models DEOS-CHAMP-01C, GGM02C and then ITG-GRACE03 appear to have better fit for medium to long wavelength and can possibly be recommended for use as long wavelength part with the local observed data. While a hybrid geopotential model selection can be achieved through the selection from either of DEOS-CHAMP-01C, GGM02C, GGM02S, EIGEN-GRACE02S or ITG-GRACE03 in the long wavelength (to degree and order 40) and EGM96, PGM200A, EIGEN-GL04C, EIGEN-CG03C or even EGM2008 in medium to short wavelength, i.e. from degree 41 to maximum degree and order.

A chemical (n, m)-graph is a connected graph of order n, size m and maximum degree at most 4. The general Randic index of a graph is defined as the sum of the weights [d(u)d(v)]α of all edges uv of the graph, where α is any real number and d(u) is the degree of a vertex u. In ...

Abstract—Fast and periodic collection of aggregated data is of considerable interest for mission-critical and continuous monitoring applications in sensor networks. In the many- to-one communication paradigm, known as convergecast, we consider applications wherein data packets are aggregated at each hop along a tree-based routing topology and focus on maximizing the data collection rate at a sink node. The primary hindrance

... Lo'ai Ali Tawalbeh & Abidalrahman Mohammad & Adnan Abdul-Aziz ... LA Tawalbeh .A. Mohammad Computer Engineering Department, Jordan University of Science and Technology, Irbid, Jordan LA Tawalbeh e-mail: tawalbeh@just.edu.jo A. Mohammad e-mail ...

A minimum degree spanning tree of a graph G is a spanning tree of G whose maximum degree is minimum among all spanning trees of G. The minimum degree spanning tree problem (MDST) is to construct such a spanning tree of a graph. In this paper, we propose a polynomial-time algorithm for solving the MDST problem on series-parallel graphs. Our algorithm runs in linear time for series-parallel graphs with small degrees. By applying this algorithm, we also give an approximation algorithm for solving the minimum edge-ranking spanning tree problem on series-parallel graphs.

The paper proposes a method for project selection under a specific decision situation, where a final selection is guided by two aspects: (1) satisfaction of certain segmentation, policy and/or logical constraints, and (2) assurance that the individual evaluation of the projects ...