Pair Of Adjacent Vertices Research Articles

Multicolorings of Series-Parallel Graphs

Let G be a graph, and let each vertex v of G have a positive integer weight ω(v). A multicoloring of G is to assign each vertex v a set of ω(v) colors so that any pair of adjacent vertices receive disjoint sets of colors. This paper presents an algorithm to find a multicoloring of a given series-parallel graph G with the minimum number of colors in time O(nW), where n is the number of vertices and W is the maximum weight of vertices in G.

Graceful Labeling of Balanced Stars of Paths

Abstract If the vertices of a tree on n vertices are labeled by the distinct numbers l,2,…, n , in such a way that the set of edge-differences (i.e., the differences between the labels on adjacent pairs of vertices) is exactly {1, 2,…, n − 1}, then such a labeling is called a graceful labeling. The graceful tree conjecture of Ringel and Kotzig which states “every tree has a graceful labeling” still remains unresolved; only the graceful labeling of some particular classes of trees are known so far. [2] is an excellent survey on the state of knowledge in this field. The main result of this paper gives graceful labeling to a class of trees called balanced stars of paths , which are defined below.

Two early branching indices and the relation between them

In the 1970s two different structure-descriptors were put forward, aimed at quantifying the extent of branching of the carbon-atom skeleton of organic molecules: Randic's connectivity index, χ=∑(δrδ2)−1/2, where δr is the degree of the vertex r of the molecular graph and where the summation goes over all pairs of adjacent vertices (1975), and the greatest eigenvalue, λ1, of the molecular graph (1973, 1977). Curiously, these two branching indices were never compared. By studying the relation between λ1 and an auxiliary quantity ρ=∑(δrδ2)+1/2, as well as the relation between ρ and χ, we establish the actual relation between χ and λ1. For differently branched isomers, there is a (rough) decreasing correlation between λ1 and χ; however, within groups of similarly branched isomers the correlation between λ1 and χ increases and is nearly linear.

Arranging Solid Balls to Represent a Graph

By solid balls, we mean a set of balls in R 3 no two of which can penetrate each other. Every finite graph G can be represented by arranging solid balls in the following way: Put red balls in R 3, one for each vertex of G, and connect two red balls by a chain when they correspond to a pair of adjacent vertices of G, where a chain means a finite sequence of blue solid balls in which each consecutive balls are tangent. (We may omit the chain if the two red balls are already tangent.) The ball number b(G) of G is the minimum number of balls (red and blue) necessary to represent G. If we put the balls and chains on a table so that all balls sit on the table, then the minimum number of balls for G is denoted by b T (G). Among other things, we prove that b(K 6)=8,b(K 7)=13 and b T (K 5)=8,b T (K 6)=14. We also prove that c 1 n 3<b(K n )<c 2 n 3 log n, c 3 n 4 log n<b T (K n )<c 4 n 4.

A user authentication protocol based on the intractability of the 3-coloring problem

In this note, we consider the problem of verifying the identity of an individual involved in a two party communication activity using a well-known Zero Knowledge protocol for a computationally intractable problem. This problem is the problem of coloring the vertices of a graph using three colors so that no pair of adjacent vertices receives the same color also known as 3-coloring problem. This problem is NP-complete, i.e. ít shares with a multitude of other natural combinatorial problems the property that most likely no fast (polynomial) algorithm exists for their solution. In this note, we use randomly generated 3-colorable graphs and use the knowledge of a 3-coloring of them to authenticate individuals. We exploìt the fact that one may easily generate a random 3-colorable graph with a specific 3-coloring that only he/she knows although it is a computationally intractable problem for someone who wants to impersonate the individual to discover a 3-coloring. Therefore, knowledge of a 3-coloring of a graph provides authentication of the individual possessing this knowledge. To prove this knowledge, the individual may use an adaptation of a Zero Knowledge Interactive proof protocol for 3-coloring.

Choosing the exponent in the definition of the connectivity index

Let ?v denote the degree of the vertex v of a molecular graph G. Then the connectivity index of G is defined as C (?) = G (?,C) = ? (?u?v)?, where the summation goes over all pairs of adjacent vertices. The exponent ? is usually chosen to be equal to -1/2, but other options were considered as well, especially ?=-1. We show that whereas C(-1/2) is a suitable measure of branching of the carbon-atom skeleton of organic molecules, and thus applicable as a topological index for modeling physico-chemical properties of the respective compounds, this is not the case with C(-1). The value of ? is established, beyond which C(?) fails to correctly reflect molecular branching.

Multigraph augmentation under biconnectivity and general edge-connectivity requirements

Given an undirected multigraph G = (V, E) and a requirement function rλ: () → Z+ (where () is the set of all pairs of vertices and Z+ is the set of nonnegative integers), we consider the problem of augmenting G by the smallest number of new edges so that the local edge-connectivity and vertex-connectivity between every pair x, y ∈ V become at least rλ(x, y) and two, respectively. In this paper, we show that the problem can be solved in O(n3(m + n) log(n2/(m + n))) time, where n and m are the numbers of vertices and pairs of adjacent vertices in G, respectively. This time complexity can be improved to O((nm + n2 log n) log n), in the case of the uniform requirement rλ(x, y)= 𝓁 for all x, y ∈ V. Furthermore, for the general rλ, we show that the augmentation problem that preserves the simplicity of the resulting graph can be solved in polynomial time for any fixed 𝓁* = max{rλ(x, y) | x, y ∈ V}. © 2001 John Wiley & Sons, Inc.

Contractible Non-Edges in Triangle-Free Graphs

Thomassen has proved that every triangle-free k-connected graph contains a pair of adjacent vertices whose identification yields again a k-connected graph. We study the existence of a pair of nonadjacent vertices whose identification preserves k-connectivity. In particular, we present a reduction theorem for the class of all bipartite, k-connected graphs.

Problems of distance geometry and convex properties of quadratic maps

A weighted graph is calledd-realizable if its vertices can be chosen ind-dimensional Euclidean space so that the Euclidean distance between every pair of adjacent vertices is equal to the prescribed weight. We prove that if a weighted graph withk edges isd-realizable for somed, then it isd-realizable for % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaaceiGaa8hzaiabg2da9maadmaabaWaaeWaaeaadaGc% aaqaaiabiIda4iaa-TgacqGHRaWkcqaIXaqmaSqabaGccqGHsislcq% aIXaqmaiaawIcacaGLPaaacqGGVaWlcqaIYaGmaiaawUfacaGLDbaa% aaa!47D5! $$d = \left[ {\left( {\sqrt {8k + 1} - 1} \right)/2} \right]$$ (this bound is sharp in the worst case). We prove that for a graphG withn vertices andk edges and for a dimensiond the image of the so-called rigidity map ℝ dn →ℝ k is a convex set in ℝ k provided % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaaceiGaa8hzaiabgwMiZoaadmaabaWaaeWaaeaadaGc% aaqaaiabiIda4iaa-TgacqGHRaWkcqaIXaqmaSqabaGccqGHsislcq% aIXaqmaiaawIcacaGLPaaacqGGVaWlcqaIYaGmaiaawUfacaGLDbaa% aaa!4895! $$d \geqslant \left[ {\left( {\sqrt {8k + 1} - 1} \right)/2} \right]$$ . These results are obtained as corollaries of a general convexity theorem for quadratic maps which also extends the Toeplitz-Hausdorff theorem. The main ingredients of the proof are the duality for linear programming in the space of quadratic forms and the “corank formula” for the strata of singular quadratic forms.

Triangles with restricted degree sum of their boundary vertices in plane graphs

It is known that under appropriate assumptions, each plane graph contains a vertex of degree at most 5 and a pair of adjacent vertices with degree sum at most 13. Two structural assumptions are established for a plane graph which together guarantee the existence of a triple of pairwise adjacent vertices with restricted degree sum. As shown by constructions, if any of these assumptions is violated, the degree sum of each three pairwise adjacent vertices may be arbitrarily large. As for a quadruple of pairwise adjacent vertices, it can hardly be forced in a plane graph by means of any reasonable restrictions.

T-chromatic polynomials

Given a set T of nonnegative integers, a T-coloring of a graph G is a labeling of the vertices of G with positive integers such that no pair of adjacent vertices is labeled with integers differing by a number in T. Let T G (λ) denote the number of ways to T-color G with numbers from the set {1, 2,…,λ}. We show that there is a polynomial, Q G (λ), such that Q G (λ) = T G (λ) provided that λ is big enough.

Procedural interpolation with geometrically continuous cubic splines

A local interpolation method for curves in R 2 or R 3 offering G 1 continuity is described. A curve is represented as a union of geometrically continuous cubic Bézier segments between each pair of adjacent vertices. At each interpolation point, the procedure determines a tangent direction and two derivative magnitudes on either side of the vertex. The method uses an intuitive geometric, rule-based approach to find a ‘good’ default solution that produces pleasing-looking results even for highly irregular sets of data Various spline properties and their relevance to the method are also discussed.

Scheduling real-time computations with separation constraints

Given a graph and a positive integer k, the Separation Problem is to find a linear layout for the vertices such that each pair of adjacent vertices has a distance of at least k in the layout. A polynomial time algorithm has been studied earlier for the special case when the graph is a directed forest. In this paper, we study the problem when the roots of the trees in the forest have different constraints on their earliest starting positions. We present an O( n log n) algorithm for the problem with n vertices in the forest. We then use the algorithm to schedule real-time jobs with minimum distance constraints.

On bridged graphs and cop-win graphs

A subgraph H of a graph G is isometric if the distance between any pair of vertices in H is the same as that in G. A graph is bridged if it contains no isometric cycles of length greater than three (roughly, each cycle of length greater than three has a shortcut). It is proved that every nontrivial bridged graph has a pair of adjacent vertices u, v, with v adjacent to everything to which u is adjacent. (This result was conjectured by P. Hell, and independently raised as a question, though in a different form, by R. E. Jamison). From this, it follows that every bridged graph G has a vertex u such that G − u is an isometric subgraph, and hence also bridged. The latter is a result of Farber. It is also proved that a connected graph is bridged if and only if every isometric subgraph is a cop-win graph, as defined by Nowakowski and Winkler.

Random walks on graphs

In this paper the following Markov chains are considered: the state space is the set of vertices of a connected graph, and for each vertex the transition is always to an adjacent vertex, such that each of the adjacent vertices has the same probability. Detailed results are given on the expectation of recurrence times, of first-entrance times, and of symmetrized first-entrance times (called commuting times). The problem of characterizing all connected graphs for which the commuting time is constant over all pairs of adjacent vertices is solved almost completely.

On the Line Graph of the Complete Bipartite Graph

In an interesting recent article [4], J. W. Moon has given a list of properties of the graph $L(B_{mn})$ (to be defined more precisely below) and investigated the question of whether these properties characterize the graph. In case $m = n$, this question had been settled by Shrikhande [5] (see also [1], [2] and [3]), who proved that the answer was yes unless $m = n = 4$, when there is exactly one exception. In case $m > n$, Moon shows that the answer is yes unless $(m, n) = (5, 4)$ or (4, 3), which cases were left unsettled in [4]. The purpose of this note is to show that, in those cases as well, the answer is yes, thus completing Moon's discussion. Now to define our problem more exactly. The line graph of the complete bipartite graph on sets with $m$ and $n$ vertices, denoted by $L(B_{mn})$, is the graph with $mn$ vertices given by all ordered pairs $(i, j), 1 \leqq i \leqq m, 1 \leqq j \leqq n$. Two vertices $(i, j)$ and $(i', j')$ are joined by an edge if $i = i'$ or $j = j'$, but not both. The graph $L(B_{mn})$ has the following properties: (1.1) It has $mn$ vertices. (1.2) Each vertex has valence $m + n - 2$. (1.3) If two vertices are not adjacent, there are exactly two vertices adjacent to each. (1.4) Of the $\frac{1}{2} mn(m + n - 2)$ pairs of adjacent vertices, exactly $n\binom{m}{2}$ pairs are each adjacent to exactly $m - 2$ vertices, the remaining $m\binom{n}{2}$ pairs are each adjacent to exactly $n - 2$ vertices. We now assume $m > n$, and let $G_{mn}$ be any graph satisfying (1.1)-(1.4). Our object is to prove that when $(m, n) = (5, 4)$ or (4, 3), $G_{mn} = L(B_{mn})$. Moon has established $G_{mn} = L(B_{mn})$ in all other cases.