Monochromatic disconnection of graphs

Abstract

For an edge-colored graph $G$, we call an edge–cut $M$ of $G$ monochromatic if the edges of $M$ are colored with the same color. The graph $G$ is called monochromatically disconnected if any two distinct vertices of $G$ are separated by a monochromatic edge–cut. For a connected graph $G$, the monochromatic disconnection number of $G$, denoted by $md\left(G\right)$, is the maximum number of colors that are needed in order to make $G$ monochromatically disconnected. We show that almost all graphs have monochromatic disconnection numbers equal to 1. We also obtain the Nordhaus–Gaddum-type results for $md\left(G\right)$.

Introduction

Let $G$ be a graph and let $V\left(G\right)$, $E\left(G\right)$ denote the vertex set and the edge set of $G$, respectively. Let $\left|G\right|$ (also $v\left(G\right)$) denote the number of vertices of $G$, called the order of $G$, and let $\Vert G\Vert$ (also $e\left(G\right)$) denote the number of edges of $G$, called the size of $G$. If there is no confusion, we use $n$ and $m$ to denote, respectively, the number of vertices and the number of edges of a graph, throughout this paper. For $v\in V\left(G\right)$, let $d_G\left(v\right)$ denote the degree of $v$. We call a vertex $v$ a $t$-degree vertex of $G$ if $d_G\left(v\right)=t$. Let $\delta \left(G\right)$ and $\Delta \left(G\right)$ denote the minimum and maximum degree of $G$, respectively. Sometimes, we also use $\Delta$ to denote a triangle. We use $\overline{G}$ to denote the complement graph of $G$. Let $S$ and $F$ be a vertex subset and an edge subset of $G$, respectively. $G-S$ is a graph obtained from $G$ by deleting the vertices of $S$ together with the edges incident with the vertices of $S$. $G-F$ is a graph whose vertex set is $V\left(G\right)$ and edge set is $E\left(G\right)-F$. Let $G\left[S\right]$ and $G\left[F\right]$ be the vertex-induced and edge-induced subgraphs of $G$, by $S$ and $F$, respectively. The distance of $u,v$ in $G$ is denoted by $d_G(u,v)$. For all other terminology and notation not defined here we follow Bondy and Murty [3].

Throughout this paper, let $K_n^{-}$ be the graph obtained from $K_n$ by deleting an arbitrary edge. $K_3$ is also called a triangle. We call a cycle $C$ a $t$-cycle if $\left|C\right|=t$. If $r$ is a positive integers, we use $\left[r\right]$ to denote the set $\{1,2,\dots ,r\}$ of integers. If $r=0$, $\left[r\right]$ is an empty set.

For a graph $G$, let $\Gamma :E\left(G\right)\to \left[r\right]$ be an edge-coloring of $G$ that allows the same color to be assigned to adjacent edges. For an edge $e$ of $G$, we use $\Gamma \left(e\right)$ to denote the color of $e$. If $H$ is a subgraph of $G$, we also use $\Gamma \left(H\right)$ to denote the set of colors on the edges of $H$ and use $\left|\Gamma \left(H\right)\right|$ to denote the number of colors in $\Gamma \left(H\right)$. An edge-coloring $\Gamma$ of $G$ is trivial if $\left|\Gamma \left(G\right)\right|=1$, otherwise, it is nontrivial.

The notion rainbow connection coloring was introduced by Chartrand et al. in [7]. A rainbow connection coloring of a graph $G$ is an edge-coloring of $G$ such that any two distinct vertices are connected by a rainbow path (a path of $G$ whose edges are colored pairwise differently.) The notion rainbow disconnection coloring was introduced by Chartrand et al. in [6]. A rainbow disconnection coloring of a graph $G$ is an edge-coloring of $G$ such that any two distinct vertices are separated by a rainbow cut (a cut of $G$ whose edges are colored pairwise differently).

Contrary to the concepts for rainbow connection and disconnection, monochromatic versions of these concepts naturally appeared, as the other extremal. A monochromatic connection coloring of a graph $G$, which was introduced by Caro and Yuster in [4], is an edge-coloring of $G$ such that any two vertices are connected by a monochromatic path (a path of $G$ whose edges are colored the same).

As a counterpart of the rainbow disconnection coloring and a similar object of the monochromatic connection coloring, we now introduce the notion of monochromatic disconnection coloring of a graph. For an edge-colored graph $G$, we call an edge–cut $M$ a monochromatic edge–cut if the edges of $M$ are colored with the same color. For two distinct vertices $u,v$ of $G$, a monochromatic $uv$-cut is a monochromatic edge–cut that separates $u$ and $v$. An edge-colored graph $G$ is monochromatically disconnected if any two distinct vertices of $G$ has a monochromatic cut separating them. An edge-coloring of $G$ is a monochromatic disconnection coloring ($MD$-coloring for short) if it makes $G$ monochromatically disconnected. For a connected graph $G$, the monochromatic disconnection number of $G$, denoted by $md\left(G\right)$, is the maximum number of colors that are needed in order to make $G$ monochromatically disconnected. An extremal MD-coloring of $G$ is an $MD$-coloring that uses $md\left(G\right)$ colors. If $H$ is a subgraph of $G$ and $\Gamma$ is an edge-coloring of $G$, we call $\Gamma$ an edge-coloring restricted on $H$.

As we know that there are two ways to study the connectivity of a graph, one way is by using paths and the other is by using cuts. Both rainbow connection and monochromatic connection provide ways to study the colored connectivity of a graph by colored paths. However, both rainbow disconnection and monochromatic disconnection can provide ways to study the colored connectivity of a graph by colored cuts. All these parameters or numbers coming from studying the colored connectivity of a graph should be regarded as some kinds of chromatic numbers. However, they are different from classic chromatic numbers. These kinds of chromatic numbers come from colorings by keeping some global structural properties of a graph, say connectivity; whereas the classic chromatic numbers come from colorings by keeping some local structural properties of a graph, say adjacent vertices or edges. So, the employed methods to study them appear quite different sometimes. Of course, local structural properties may yield global structural properties, and vice versa. But this is not always the case, say, local connectedness of a graph cannot guarantee connectedness of the entire graph. So, many colored versions of connectivity parameters appeared in recent years, and we refer the reader to [2], [9], [10], [11], [13], [14], [15], [16] for surveys.

Let $G$ be a graph that may have parallel edges but no loops. By deleting all parallel edges but one of them, we obtain a simple spanning subgraph of $G$, and call it the underling graph of $G$. If there are some parallel edges of an edge $e=ab$, then any monochromatic $ab$-cut contains $e$ and its parallel edges. Therefore, the following result is obvious, which means that we only need to think about simple graphs in the sequel.

Proposition 1.1

Let $G^{\prime }$ be the underling graph of a graph $G$. Then $md\left(G\right)=md\left(G^{\prime }\right)$.

The following result means that we only need to consider connected graphs in the sequel.

Proposition 1.2

If a simple graph $G$ has $t$ components $D_1,\dots ,D_t$, then $md\left(G\right)={\sum }_{i\in \left[t\right]}md\left(D_i\right)$.

Let $G$ and $H$ be two graphs. The union of $G$ and $H$ is the graph $G\cup H$ with vertex set $V\left(G\right)\cup V\left(H\right)$ and edge set $E\left(G\right)\cup E\left(H\right)$. If $G$ and $H$ are vertex-disjoint, then let $G\vee H$ denote the join of $G$ and $H$, which is obtained from $G$ and $H$ by adding an edge between each vertex of $G$ and every vertex of $H$.

A block of a graph $G$ is trivial if it is a cut–edge of $G$. If $e=uv$ is an edge of $G$ with $d_G\left(v\right)=1$, we call $e$ a pendent edge of $G$ and $v$ a pendent vertex of $G$.