Rainbow monochromatic $k$-edge-connection colorings of graphs

TL;DR

This paper introduces rainbow monochromatic $k$-edge-connection colorings in graphs, establishing existence, bounds, and thresholds for the maximum number of colors ensuring such colorings, with implications for graph connectivity.

Contribution

The paper defines the concept of rainbow monochromatic $k$-edge-connection colorings, proves their existence, derives bounds for the maximum number of colors, and determines threshold functions for random graphs.

Findings

Existence of $RMC_k$-colorings in graphs.

Bounds for the rainbow monochromatic $k$-edge-connection number.

Threshold function for $rmc_k(G(n,p))$ in random graphs.

Abstract

A path in an edge-colored graph is called a monochromatic path if all edges of the path have a same color. We call $k$ paths $P_1,\cdots,P_k$ rainbow monochromatic paths if every $P_i$ is monochromatic and for any two $i\neq j$, $P_i$ and $P_j$ have different colors. An edge-coloring of a graph $G$ is said to be a rainbow monochromatic $k$-edge-connection coloring (or $RMC_k$-coloring for short) if every two distinct vertices of $G$ are connected by at least $k$ rainbow monochromatic paths. We use $rmc_k(G)$ to denote the maximum number of colors that ensures $G$ has an $RMC_k$-coloring, and this number is called the rainbow monochromatic $k$-edge-connection number. We prove the existence of $RMC_k$-colorings of graphs, and then give some bounds of $rmc_k(G)$ and present some graphs whose $rmc_k(G)$ reaches the lower bound. We also obtain the threshold function for $rmc_k(G(n,p))\geq f(n)$, where $\lfloor\frac{n}{2}\rfloor> k\geq 1$.