[Paper Review] Monochromatic disconnection: Erdős-Gallai-type problems and product graphs

Ping Li, Xueliang Li|arXiv (Cornell University)|Apr 18, 2019

Advanced Graph Theory Research11 references4 citations

TL;DR

This paper investigates the monochromatic disconnection number $md(G)$, defined as the maximum number of colors in an edge-coloring that separates every pair of vertices via a monochromatic edge-cut. It resolves Erdős-Gallai-type extremal problems for this invariant and derives exact values of $md(G)$ for four graph products—Cartesian, strong, lexicographic, and tensor products—showing that $md(G \ast H) \leq \min\{g_o(G), g_o(H)\}$ when at least one factor is non-bipartite and has no pendant edges.

ABSTRACT

For an edge-colored graph $G$, we call an edge-cut $M$ of $G$ monochromatic if the edges of $M$ are colored with a same color. The graph $G$ is called monochromatically disconnected if any two distinct vertices of $G$ are separated by a monochromatic edge-cut. The monochromatic disconnection number, denoted by $md(G)$, of a connected graph $G$ is the maximum number of colors that are allowed to make $G$ monochromatically disconnected. In this paper, we solve the Erdős-Gallai-type problems for the monochromatic disconnection, and give the monochromatic disconnection numbers for four graph products, i.e., Cartesian, strong, lexicographic, and tensor products.

Motivation & Objective

• To characterize the maximum number of colors in an edge-coloring that ensures every pair of vertices is separated by a monochromatic edge-cut.
• To resolve Erdős-Gallai-type extremal problems for the monochromatic disconnection number $md(G)$, identifying extremal graphs that maximize or minimize $md(G)$ under given constraints.
• To determine exact values of $md(G)$ for four standard graph products: Cartesian, strong, lexicographic, and tensor products.
• To establish structural conditions under which $md(G \ast H) = 1$, particularly when one factor contains a triangle or is a complete graph $K_n$ with $n \geq 5$.
• To generalize bounds on $md(G \ast H)$ using the odd girth of non-bipartite factors, providing tight upper limits based on cycle structure.

Proposed method

• Uses the concept of monochromatic edge-cuts—edge-cuts where all edges have the same color—to define monochromatic disconnection and the monochromatic disconnection number $md(G)$.
• Applies structural graph theory, including block decomposition and spanning subgraphs, to reduce problems to simpler components like trees and cycles.
• Employs induction and subgraph restriction techniques to bound $md(G \ast H)$, particularly by analyzing $md(G' \ast H)$ for subgraphs $G'$ of $G$.
• Leverages the odd girth $g_o(G)$ of a non-bipartite graph $G$ as a key parameter to bound $md(G \ast H)$, showing $md(G \ast H) \leq \min\{g_o(G), g_o(H)\}$.
• Uses known results on $md(K_n)$, $md(K_n^-)$, and $md(K_{n,t})$ to derive $md(G \ast H) = 1$ when one factor is a complete graph $K_n$ with $n \geq 5$.
• Applies the concept of soft-layers and vertex sequences in product graphs to control the structure of color-induced edge sets and verify $MD$-colorings.

Experimental results

Research questions

• RQ1 What is the maximum number of colors $md(G)$ that can be used in an edge-coloring of a connected graph $G$ such that every pair of vertices is separated by a monochromatic edge-cut?
• RQ2 Which graphs achieve the extremal values of $md(G)$ under constraints such as number of vertices, minimum degree, or cycle structure?
• RQ3 How does the monochromatic disconnection number behave under the four standard graph products: Cartesian, strong, lexicographic, and tensor products?
• RQ4 Under what conditions does $md(G \ast H) = 1$ for graph products, particularly when one factor is a complete graph or contains a triangle?
• RQ5 Can the monochromatic disconnection number of a product graph be bounded in terms of the odd girth of its factors?

Key findings

• The monochromatic disconnection number $md(G)$ satisfies $1 \leq md(G) \leq n-1$ for a connected graph $G$ of order $n$, with equality to $n-1$ if and only if $G$ is a tree.
• For a unicyclic graph $G$ with cycle length $n$, $md(G) = \left\lfloor \frac{n}{2} \right\rfloor$ if $G$ is not a triangle, and $md(G) \leq n-2$ otherwise.
• For the Cartesian product $G \Box H$, if both $G$ and $H$ are non-bipartite and have no pendant edges, then $md(G \Box H) \leq \min\{g_o(G), g_o(H)\}$.
• For the tensor product $G \times H$, if $G$ and $H$ are both non-bipartite and have minimum degree at least 2, then $md(G \times H) \leq \min\{g_o(G), g_o(H)\}$.
• If $H = K_n$ with $n \geq 5$, then $md(G \times H) = 1$ for any connected graph $G$ with $|G| \geq 2$, due to the high connectivity and odd girth of $K_n$.
• If $G$ is neither a tree nor a unicyclic graph with a $K_3$ cycle, and $H$ contains a triangle with no pendant edges, then $md(G \times H) = 1$.