Rainbow Connection Number and Connected Dominating Sets

Rainbow connection number rc(G) of a connected graph G is the minimum number of colours needed to colour the edges of G, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we show that for every connected graph G, with minimum degree at least 2, the rainbow connection number is upper bounded by {\gamma}_c(G) + 2, where {\gamma}_c(G) is the connected domination number of G. Bounds of the form diameter(G) \leq rc(G) \leq diameter(G) + c, 1 \leq c \leq 4, for many special graph classes follow as easy corollaries from this result. This includes interval graphs, AT-free graphs, circular arc graphs, threshold graphs, and chain graphs all with minimum degree at least 2 and connected. We also show that every bridge-less chordal graph G has rc(G) \leq 3.radius(G). In most of these cases, we also demonstrate the tightness of the bounds. An extension of this idea to two-step dominating sets is used to show that for every connected graph on n vertices with minimum degree {\delta}, the rainbow connection number is upper bounded by 3n/({\delta} + 1) + 3. This solves an open problem of Schiermeyer (2009), improving the previously best known bound of 20n/{\delta} by Krivelevich and Yuster (2010). Moreover, this bound is seen to be tight up to additive factors by a construction of Caro et al. (2008).Comment: 14 page

Rainbow Connection Number and Radius

The rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of its vertices is connected by at least one path in which no two edges are coloured the same. In this note we show that for every bridgeless graph G with radius r, rc(G) <= r(r + 2). We demonstrate that this bound is the best possible for rc(G) as a function of r, not just for bridgeless graphs, but also for graphs of any stronger connectivity. It may be noted that for a general 1-connected graph G, rc(G) can be arbitrarily larger than its radius (Star graph for instance). We further show that for every bridgeless graph G with radius r and chordality (size of a largest induced cycle) k, rc(G) <= rk. It is known that computing rc(G) is NP-Hard [Chakraborty et al., 2009]. Here, we present a (r+3)-factor approximation algorithm which runs in O(nm) time and a (d+3)-factor approximation algorithm which runs in O(dm) time to rainbow colour any connected graph G on n vertices, with m edges, diameter d and radius r.Comment: Revised preprint with an extra section on an approximation algorithm. arXiv admin note: text overlap with arXiv:1101.574

Minimum Breadth of a Graph

Breadth of a graph as the maximum of heights taken over all diametral paths is investigated in [3, 4], where height is taken by placing each diametral path on level y = 0 and placing uniquely the rest of the vertices on levels y = 1, 2…k keeping adjacency intact. A parameter minimum breadth is introduced as minimum of heights with respect to all diametral paths. A few results on minimum breadth in certain classes of graphs are presented. Also the bounds on number of vertices and edges for graphs of known diameter and minimum breadth are proposed

Broadcast CONGEST Algorithms against Adversarial Edges

We consider the corner-stone broadcast task with an adaptive adversary that controls a fixed number of $t$ edges in the input communication graph. In this model, the adversary sees the entire communication in the network and the random coins of the nodes, while maliciously manipulating the messages sent through a set of $t$ edges (unknown to the nodes). Since the influential work of [Pease, Shostak and Lamport, JACM'80], broadcast algorithms against plentiful adversarial models have been studied in both theory and practice for over more than four decades. Despite this extensive research, there is no round efficient broadcast algorithm for general graphs in the CONGEST model of distributed computing. We provide the first round-efficient broadcast algorithms against adaptive edge adversaries. Our two key results for $n$-node graphs of diameter $D$ are as follows: 1. For $t=1$, there is a deterministic algorithm that solves the problem within $\widetilde{O}(D^2)$ rounds, provided that the graph is 3 edge-connected. This round complexity beats the natural barrier of $O(D^3)$ rounds, the existential lower bound on the maximal length of $3$ edge-disjoint paths between a given pair of nodes in $G$. This algorithm can be extended to a $\widetilde{O}(D^{O(t)})$-round algorithm against $t$ adversarial edges in $(2t+1)$ edge-connected graphs. 2. For expander graphs with minimum degree of $\Omega(t^2\log n)$, there is an improved broadcast algorithm with $O(t \log ^2 n)$ rounds against $t$ adversarial edges. This algorithm exploits the connectivity and conductance properties of G-subgraphs obtained by employing the Karger's edge sampling technique. Our algorithms mark a new connection between the areas of fault-tolerant network design and reliable distributed communication.Comment: accepted to DISC2

Diameter of 3-Colorable Graphs and Some Remarks on the Midrange Crossing Constant

The first part of this dissertation discussing the problem of bounding the diameter of a graph in terms of its order and minimum degree. The initial problem was solved independently by several authors between 1965 − 1989. They proved that for fixed δ ≥ 2 and large n, diam(G) ≤ 3n+ O(1). In 1989, Erdős, Pach, Pollack, and Tuza conjectured that the upper bound on the diameter can be improved if G does not contain a large complete subgraph Kk. Let r, δ ≥ 2 be fixed integers and let G be a connected graph with n vertices and minimum degree δ. In general, Erdős et al. conjectured tight upper bounds for K2r -free and K2r+1-free graphs that are better than the known δ+1 + O(1). Particular to this dissertation, their conjecture stated that K5-free graphs with 5 l δ have diameter ≤ 2δ + O(1), while K4-free graphs with 8 l δ have diameter ≤ 7δ + O(1). The first progress towards this conjecture was published by Czabarka, Dankel- mann, and Székely in 2008. They worked under the stronger assumption for when r = 2, that the graphs are 4-colorable rather than K5-free. They showed that for every connected 4-colorable graph G of order n and δ ≥ 1, diam(G) ≤ 5n − 1. We provide a counterexample to this 30 years old unsolved conjecture for K4-free graphs by showing classes of 3-colorable graphs with diameter 7n 3δ+3+ O(1). From here we conjectured that 3-colorable graphs has diameter at most 7n + O(1). We use the Duality of Linear Programming to prove 3-colorable graphs have diameter at most 2δ + O(1). We then utilize inclusion-exclusion into a different linear programming approach to prove a smaller upper bound that for every connected 3-colorable graph G of order n and δ ≥ 1, diam(G) ≤ 189n + O (1). The second part of this dissertation gives some remarks on the midrange crossing constant. The celebrated Crossing Lemma states that for any graph on n vertices and m C 4n edges we have cr(G) n2 m3 is at least 1 64 . A decade before the Crossing Lemma, Erdos and Guy made the bold conjecture that, if we denote by k(n,m) the minimum crossing number of n-vertex graph with at least m = m(n) edges, then there is a positive constant , dubbed as the midrange crossing constant, such that = lim n ∞ ª k(n,m) n2 m3 as long as m is both superlinear in n. Pach, Spencer and Tóth showed that the Erdos-Guy conjecture is true with the additional (and needed) assumption that m is subquadratic. Pach, Radoicic, Tardos and Tóth gave a construction yielding B 8 9 ϖ 2 ≈0.0900633 for the rectilinear midrange crossing constant. Details of neither of these calculations, which are said to be long and unpleasant, are available to the public. We provide a simple alternative construction that yields the same upper bound