Proper conflict-free coloring of Mycielskians with fast algorithms

Abstract

A proper conflict-free coloring, often termed as PCF-coloring, of a graph refers to a proper vertex coloring wherein each vertex's open neighborhood contains at least one color appearing exactly once. PCF-coloring boasts a wide range of applications, from theoretical graph analysis to practical applications in networking, geographic information systems, scheduling, and constraint satisfaction problems. Caro, Petruševski, and Škrekovski conjectured in 2023 that every graph with maximum degree $\Delta \ge 3$ has a PCF-$(\Delta +1)$-coloring. In this paper, we validate this conjecture for the Mycielskians of 1-subdivided graphs. Additionally, we determine the PCF-chromatic number for Mycielskians of paths, cycles, complete graphs, complete bipartite graphs, and wheels, and provide efficient algorithms that produce optimal PCF-colorings for these graphs.