Monochromatic disconnection of graphs☆
Introduction
Let be a graph and let , denote the vertex set and the edge set of , respectively. Let (also ) denote the number of vertices of , called the order of , and let (also ) denote the number of edges of , called the size of . If there is no confusion, we use and to denote, respectively, the number of vertices and the number of edges of a graph, throughout this paper. For , let denote the degree of . We call a vertex a -degree vertex of if . Let and denote the minimum and maximum degree of , respectively. Sometimes, we also use to denote a triangle. We use to denote the complement graph of . Let and be a vertex subset and an edge subset of , respectively. is a graph obtained from by deleting the vertices of together with the edges incident with the vertices of . is a graph whose vertex set is and edge set is . Let and be the vertex-induced and edge-induced subgraphs of , by and , respectively. The distance of in is denoted by . For all other terminology and notation not defined here we follow Bondy and Murty [3].
Throughout this paper, let be the graph obtained from by deleting an arbitrary edge. is also called a triangle. We call a cycle a -cycle if . If is a positive integers, we use to denote the set of integers. If , is an empty set.
For a graph , let be an edge-coloring of that allows the same color to be assigned to adjacent edges. For an edge of , we use to denote the color of . If is a subgraph of , we also use to denote the set of colors on the edges of and use to denote the number of colors in . An edge-coloring of is trivial if , otherwise, it is nontrivial.
The notion rainbow connection coloring was introduced by Chartrand et al. in [7]. A rainbow connection coloring of a graph is an edge-coloring of such that any two distinct vertices are connected by a rainbow path (a path of 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 is an edge-coloring of such that any two distinct vertices are separated by a rainbow cut (a cut of 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 , which was introduced by Caro and Yuster in [4], is an edge-coloring of such that any two vertices are connected by a monochromatic path (a path of 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 , we call an edge–cut a monochromatic edge–cut if the edges of are colored with the same color. For two distinct vertices of , a monochromatic -cut is a monochromatic edge–cut that separates and . An edge-colored graph is monochromatically disconnected if any two distinct vertices of has a monochromatic cut separating them. An edge-coloring of is a monochromatic disconnection coloring (-coloring for short) if it makes monochromatically disconnected. For a connected graph , the monochromatic disconnection number of , denoted by , is the maximum number of colors that are needed in order to make monochromatically disconnected. An extremal MD-coloring of is an -coloring that uses colors. If is a subgraph of and is an edge-coloring of , we call an edge-coloring restricted on .
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 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 , and call it the underling graph of . If there are some parallel edges of an edge , then any monochromatic -cut contains 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 be the underling graph of a graph . Then .
The following result means that we only need to consider connected graphs in the sequel.
Proposition 1.2 If a simple graph has components , then .
Let and be two graphs. The union of and is the graph with vertex set and edge set . If and are vertex-disjoint, then let denote the join of and , which is obtained from and by adding an edge between each vertex of and every vertex of .
A block of a graph is trivial if it is a cut–edge of . If is an edge of with , we call a pendent edge of and a pendent vertex of .
Section snippets
Some basic results
Let be a graph with at least two blocks. An edge-coloring of is an -coloring if and only if it is also an -coloring restricted on each block of . Therefore, the following result is obvious.
Remark 2.1 If a connected graph has blocks , then .
From the above remark, if is a tree, then .
Proposition 2.2 If is a cycle, then . Furthermore, if is a unicycle graph with cycle , then .
Proof From Remark 2.1, we only need to prove that if is a cycle.
Graphs with monochromatic disconnection number one
In this section we consider the monochromatic disconnection numbers for some special graphs, such as triangular graphs (i.e., graphs with each of its edges in a triangle), complete multipartite graphs, chordal graphs, square graphs and line graphs (the definitions of the last four graphs are well-known, we omit them). We denote the square graph and the line graph of a graph by and , respectively.
For a graph , we define a relation on the edge set as follows: for two edges and
Nordhaus–Gaddum-type results
For a graph parameter, it is always interesting to get the Nordhaus–Gaddum-type results, see [1] and [5], [8], [12], [17], [18], [19], [20] for more such results on various kinds of graph parameters. This section is devoted to get the Nordhaus–Gaddum-type results for our parameter .
For a connected graph , a vertex is deletable if is connected. Let be the set of blocks of and be the set of cut–vertices of . A block–tree of is a bipartite graph with bipartition and ,
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors are very grateful to the reviewers and editor for their very useful suggestions and comments, which helped to improving the presentation of the paper.
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