A. Manas | Searcharxiv

arXiv2026

How many colours to connect cliques?

We study the following problem: given an edge-colouring of a $K_n$ with $r$ colours, how many colours does one need to keep (in the worst-case scenario) so that the subgraph formed by those colours is connected? We determine this extremal function up to constant factors in every parameter regime, and show that it has the same asymptotic order as the corresponding number (of colours) needed for ensuring that no vertex is isolated. This gives a complete description of the ``spanning'' threshold for connectivity in edge-coloured complete graphs.

math.CO↗

arXiv2025

Sorting with constraints

In this work, we study the generalized sorting problem, where we are given a set of $n$ elements to be sorted, but only a subset of all possible pairwise element comparisons is allowed. We look at the problem from the perspective of the graph formed by the ``forbidden'' pairs, and we parameterize algorithms using the clique number and the chromatic number of this graph. We also extend these results to the class of problems where the input graph is not necessarily sortable, and one is only interested in discovering the partial order. We use our results to develop a simple algorithm that always determines the underlying partial order in $O(n^{3/2} \log n)$ probes, when the input graph is an Erd\H{o}s--R\'enyi graph.

cs.DS↗