Bipartite powers of k-chordal graphs

Let k be an integer and k \geq 3. A graph G is k-chordal if G does not have an induced cycle of length greater than k. From the definition it is clear that 3-chordal graphs are precisely the class of chordal graphs. Duchet proved that, for every positive integer m, if G^m is chordal then so is G^{m+2}. Brandst\"adt et al. in [Andreas Brandst\"adt, Van Bang Le, and Thomas Szymczak. Duchet-type theorems for powers of HHD-free graphs. Discrete Mathematics, 177(1-3):9-16, 1997.] showed that if G^m is k-chordal, then so is G^{m+2}. Powering a bipartite graph does not preserve its bipartitedness. In order to preserve the bipartitedness of a bipartite graph while powering Chandran et al. introduced the notion of bipartite powering. This notion was introduced to aid their study of boxicity of chordal bipartite graphs. Given a bipartite graph G and an odd positive integer m, we define the graph G^{[m]} to be a bipartite graph with V(G^{[m]})=V(G) and E(G^{[m]})={(u,v) | u,v \in V(G), d_G(u,v) is odd, and d_G(u,v) \leq m}. The graph G^{[m]} is called the m-th bipartite power of G. In this paper we show that, given a bipartite graph G, if G is k-chordal then so is G^{[m]}, where k, m are positive integers such that k \geq 4 and m is odd.Comment: 10 page

On Rainbow Connection Number and Connectivity

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 vertices is connected by at least one path in which no two edges are coloured the same. In this paper we investigate the relationship of rainbow connection number with vertex and edge connectivity. It is already known that for a connected graph with minimum degree $\delta$, the rainbow connection number is upper bounded by $3n/(\delta + 1) + 3$ [Chandran et al., 2010]. This directly gives an upper bound of $3n/(\lambda + 1) + 3$ and $3n/(\kappa + 1) + 3$ for rainbow connection number where $\lambda$ and $\kappa$, respectively, denote the edge and vertex connectivity of the graph. We show that the above bound in terms of edge connectivity is tight up-to additive constants and show that the bound in terms of vertex connectivity can be improved to $(2 + \epsilon)n/\kappa + 23/ \epsilon^2$, for any $\epsilon > 0$. We conjecture that rainbow connection number is upper bounded by $n/\kappa + O(1)$ and show that it is true for $\kappa = 2$. We also show that the conjecture is true for chordal graphs and graphs of girth at least 7.Comment: 10 page

Boxicity of Line Graphs

Boxicity of a graph H, denoted by box(H), is the minimum integer k such that H is an intersection graph of axis-parallel k-dimensional boxes in R^k. In this paper, we show that for a line graph G of a multigraph, box(G) <= 2\Delta(\lceil log_2(log_2(\Delta)) \rceil + 3) + 1, where \Delta denotes the maximum degree of G. Since \Delta <= 2(\chi - 1), for any line graph G with chromatic number \chi, box(G) = O(\chi log_2(log_2(\chi))). For the d-dimensional hypercube H_d, we prove that box(H_d) >= (\lceil log_2(log_2(d)) \rceil + 1)/2. The question of finding a non-trivial lower bound for box(H_d) was left open by Chandran and Sivadasan in [L. Sunil Chandran and Naveen Sivadasan. The cubicity of Hypercube Graphs. Discrete Mathematics, 308(23):5795-5800, 2008]. The above results are consequences of bounds that we obtain for the boxicity of fully subdivided graphs (a graph which can be obtained by subdividing every edge of a graph exactly once).Comment: 14 page

IN VIVO CONTACT MECHANICS OF THE DISTAL RADIOULNAR JOINT WITH AND WITHOUT SCAPHOLUNATE DISSOCIATION

The distal radioulnar joint (DRUJ) is a joint of the wrist which allows force transmission and forearm rotation in the upper limb while preserving the stability of the forearm independent of elbow and wrist flexion and extension. DRUJ is a commonly injured part of the body. Conditions affecting the joint could be positive ulnar variance or negative ulnar variance, the length of the ulna relative to radius. It is also adversely affected by nearby injuries such as distal radial fractures. In fact, a significant correlation was found between negative ulnar variance and scapholunate dissociation (SLD), a ligament injury of the wrist. This leads to the question of whether or not SLD causes changes in the radioulnar joint mechanics. Altered joint mechanics are associated with the onset of osteoarthritis (OA). An understanding of the of the normal and pathological wrist in vivo DRUJ contact mechanics should help physicians make better clinical recommendations and improve treatment for the primary injury. Proper treatment of the DRUJ could help prevent the onset of OA. Image registration is used in our modeling to determine the kinematic transformations for carpal bones from the unloaded to the loaded configuration. A perturbation study was done to evaluate the effect of varying initial manual registrations and the relative image plane orientations on the final registration kinematics. The results of the study showed that Subject II (with different imaging plane orientations) was found to have greater translation errors compared to subject I (consistent imaging planes). This result emphasizes the need to be consistent with forearm position and/or image plane orientation to minimize the errors of translation and attitude vectors. In a separate study, five additional subjects with unilateral SLD participated in another study in which MRI based contact modeling was used to analyze the contact mechanics parameters of the injured wrist compared to the normal wrist. The contact forces, peak contact pressures, average pressures and contact areas generally trended to be higher in injured wrists compared to the normal and surgically repaired wrists. Model contact areas were found to be consistent with the directly measured areas from the grasp MR images. A repeatability test was done on a single subject and the absolute differences between the contact parameters for both the trials were close. These findings suggest that SLD injury of the wrist may have an effect on the DRUJ mechanics