L. Sunil Chandran

Rainbow connection number and connected dominating sets

The rainbow connection number of a connected graph is the minimum number of colors needed to color its edges, so that every pair of its vertices is connected by at least one path in which no two edges are colored the same. In this article we show that for every connected graph on n vertices with minimum degree δ, the rainbow connection number is upper bounded by 3n/(δ + 1) + 3. This solves an open problem from Schiermeyer (Combinatorial Algorithms, Springer, Berlin/Hiedelberg, 2009, pp. 432–437), improving the previously best known bound of 20n/δ (J Graph Theory 63 (2010), 185–191). This bound is tight up to additive factors by a construction mentioned in Caro et al. (Electr J Combin 15(R57) (2008), 1). As an intermediate step we obtain an upper bound of 3n/(δ + 1) − 2 on the size of a connected two-step dominating set in a connected graph of order n and minimum degree δ. This bound is tight up to an additive constant of 2. This result may be of independent interest. We also show that for every connected graph G with minimum degree at least 2, the rainbow connection number, rc(G), is upper bounded by Γc(G) + 2, where Γc(G) is the connected domination number of G. Bounds of the form diameter(G)⩽rc(G)⩽diameter(G) + c, 1⩽c⩽4, for many special graph classes follow as easy corollaries from this result. This includes interval graphs, asteroidal triple-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)⩽3·radius(G). In most of these cases, we also demonstrate the tightness of the bounds.

Refined memorization for vertex cover

Memorization is a technique which allows to speed up exponential recursive algorithms at the cost of an exponential space complexity. This technique already leads to the currently fastest algorithm for fixed-parameter vertex cover, whose time complexity is O(1.2832^kk^1^.^5+kn), where n is the number of nodes and k is the size of the vertex cover. Via a refined use of memorization, we obtain an O(1.2759^kk^1^.^5+kn) algorithm for the same problem. We moreover show how to further reduce the complexity to O(1.2745^kk^4+kn).

Boxicity and maximum degree

A d-dimensional box is a Cartesian product of d closed intervals on the real line. The boxicity of a graph is the minimum dimension d such that it is representable as the intersection graph of d-dimensional boxes. We give a short constructive proof that every graph with maximum degree D has boxicity at most 2D^2. We also conjecture that the best upper bound is linear in D.

Girth and treewidth

The length of the shortest cycle in a graph G is called the girth of G. In particular, we show that if G has girth at least g and average degree at least d, then tw(G) = Ω(1/g+1 (d - 1)⌊(g - 1)/2⌋). In view of a famous conjecture regarding the existence of graphs with girth g, minimum degree δ and having at most c(δ - 1)⌊(g - 1)/2⌋ vertices (for some constant c), this lower bound seems to be almost tight (but for a multiplicative factor of g + 1).

A High Girth Graph Construction

We give a deterministic algorithm that constructs a graph of girth logk(n) + O(1) and minimum degree k-1, taking number of nodes n and number of edges

Boxicity and Poset Dimension

e = {\left \lfloor nk / 2 \right \rfloor }

Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes

(where

A spectral lower bound for the treewidth of a graph and its consequences

k < \frac {n}{3}

Approximations for ATSP with Parametrized Triangle Inequality

) as input. The degree of each node is guaranteed to be k-1, k, or k+1, where k is the average degree. Although constructions that achieve higher values of girth---up to

Chordal Bipartite Graphs with High Boxicity

\frac {4}{3} \log_{k-1}{(n)}