On coloring parameters of triangle-free planar (n,m)(n,m)-graphs

An $(n,m)$-graph is a graph with $n$ types of arcs and $m$ types of edges. A homomorphism of an $(n,m)$-graph $G$ to another $(n,m)$-graph $H$ is a vertex mapping that preserves the adjacencies along with their types and directions. The order of a smallest (with respect to the number of vertices) such $H$ is the $(n,m)$-chromatic number of $G$.Moreover, an $(n,m)$-relative clique $R$ of an $(n,m)$-graph $G$ is a vertex subset of $G$ for which no two distinct vertices of $R$ get identified under any homomorphism of $G$. The $(n,m)$-relative clique number of $G$, denoted by $\omega_{r(n,m)}(G)$, is the maximum $|R|$ such that $R$ is an $(n,m)$-relative clique of $G$. In practice, $(n,m)$-relative cliques are often used for establishing lower bounds of $(n,m)$-chromatic number of graph families. Generalizing an open problem posed by Sopena [Discrete Mathematics 2016] in his latest survey on oriented coloring, Chakroborty, Das, Nandi, Roy and Sen [Discrete Applied Mathematics 2022] conjectured that $\omega_{r(n,m)}(G) \leq 2 (2n+m)^2 + 2$ for any triangle-free planar $(n,m)$-graph $G$ and that this bound is tight for all $(n,m) \neq (0,1)$.In this article, we positively settle this conjecture by improving the previous upper bound of $\omega_{r(n,m)}(G) \leq 14 (2n+m)^2 + 2$ to $\omega_{r(n,m)}(G) \leq 2 (2n+m)^2 + 2$, and by finding examples of triangle-free planar graphs that achieve this bound. As a consequence of the tightness proof, we also establish a new lower bound of $2 (2n+m)^2 + 2$ for the $(n,m)$-chromatic number for the family of triangle-free planar graphs.Comment: 22 Pages, 5 figure

SS 433: Results of a Recent Multi-wavelength Campaign

We conducted a multi-wavelength campaign in September-October, 2002, to observe SS 433. We used 45 meter sized 30 dishes of Giant Meter Radio Telescope (GMRT) for radio observation, 1.2 meter Physical Research Laboratory Infra-red telescope at Mt Abu for IR, 1 meter Telescope at the State Observatory, Nainital for Optical photometry, 2.3 meter optical telescope at the Vainu Bappu observatory for spectrum and Rossi X-ray Timing Explorer (RXTE) Target of Opportunity (TOO) observation for X-ray observations. We find sharp variations in intensity in time-scales of a few minutes in X-rays, IR and radio wavelengths. Differential photometry at the IR observation clearly indicated significant intrinsic variations in short time scales of minutes throughout the campaign. Combining results of these wavelengths, we find a signature of delay of about two days between IR and Radio. The X-ray spectrum yielded double Fe line profiles which corresponded to red and blue components of the relativistic jet. We also present the broadband spectrum averaged over the campaign duration.Comment: 17 pages 10 figures MNRAS (submitted

Possible Photometric Evidence of Ejection of Bullet Like Features in the Relativistic Jet source SS433

SS433 is well-known for its precessing twin jets having optical bullets inferred through {\it spectroscopic} observation of $H_\alpha$ lines. Recently, Chakrabarti et al. (2002) described processes which may be operating in accretion disk of SS433 to produce these bullets. In a recent multi-wavelength campaign, we find sharp rise in intensity in time-scales of few minutes in X-rays, IR and radio waves through {\it photometric} studies. We interpret them to be possible evidence of ejection of bullet-like features from accretion disks.Comment: 9 latex pages with five figure

Two-center of the convex hull of a point set: Dynamic model, and restricted streaming model

In this paper, we consider the dynamic version of covering the convex hull of a point set P in R2 by two congruent disks of minimum size. Here, the points can be added or deleted in the set P, and the objective is to maintain a data structure that, at any instant of time, can efficiently report two disks of minimum size whose union completely covers the boundary of the convex hull of the point set P. We show that maintaining a linear size data structure, we can report a radius r satisfying r 2ropt at any query time, where ropt is the optimum solution at that instant of time. For each insertion or deletion of a point in P, the update time of our data structure is O(log n). Our algorithm can be tailored to work in the restricted streaming model where only insertions are allowed, using constant work-space. The problem studied in this paper has novelty in two ways: (i) it computes the covering of the convex hull of a point set P, which has lot of surveillance related applications, but not studied in the literature, and (ii) it also considers the dynamic version of the problem. In the dynamic setup, the extent measure problems are studied very little, and in particular, the k-center problem is not at all studied for any k2