Can we efficiently color interval graphs using parallel computing? This research investigates the complexity of minimally vertex coloring interval graphs, a fundamental problem in graph theory with applications in scheduling and resource allocation. The work explores the possibility that minimally vertex coloring an interval graph may not be in NC1, suggesting inherent limitations in parallelizability. This research has strong applications to **computer science** and parallel computing. It's shown that 3-coloring a linked list (a problem known to be difficult to parallelize) is NC1-reducible to minimally coloring an interval graph. However, the paper demonstrates that an interval graph with a known interval representation and a small (O(1)) chromatic number can be minimally colored in NC1. Additionally, an o(log n) time, polynomial processors algorithm is obtained for minimally coloring an interval graph with o(log n) chromatic number. The findings suggest a trade-off between the chromatic number of the graph and the efficiency of parallel coloring algorithms. While minimally coloring general interval graphs may be inherently difficult to parallelize, specific classes of interval graphs admit efficient parallel solutions. Algorithms make use of NC and PRAM structures for **computer** calculations.
Published in International Journal of Foundations of Computer Science, this paper is highly relevant to the journal's focus on the theoretical foundations of computer science. The investigation into the parallel complexity of graph coloring problems directly contributes to the journal's mission of advancing knowledge in algorithms and computational complexity. This has use for electrical engineering.