More than ten years ago in 2008, a new kind of graph coloring appeared in graph theory, which is the {\it rainbow connection coloring} of graphs, and then followed by some other new concepts of graph colorings, such as {\it proper connection coloring, monochromatic connection coloring, and conflict-free connection coloring} of graphs. In about ten years of our consistent study, we found that these new concepts of graph colorings are actually quite different from the classic graph colorings. These {\it colored connection colorings} of graphs are brand-new colorings and they need to take care of global structural properties (for example, connectivity) of a graph under the colorings; while the traditional colorings of graphs are colorings under which only local structural properties (adjacent vertices or edges) of a graph are taken care of. Both classic colorings and the new colored connection colorings can produce the so-called chromatic numbers. We call the colored connection numbers the {\it global chromatic numbers}, and the classic or traditional chromatic numbers the {\it local chromatic numbers}. This paper intends to clarify the difference between the colored connection colorings and the traditional colorings, and finally to propose the new concepts of global colorings under which global structural properties of the colored graph are kept, and the global chromatic numbers.

Graph colorings under global structural conditions.

An edge-cut $R$ of an edge-colored connected graph is called a rainbow-cut if no two edges in the edge-cut are colored the same. An edge-colored graph is rainbow disconnected if for any two distinct vertices $u$ and $v$ of the graph, there exists a $u$-$v$-rainbow-cut separating them. For a connected graph $G$, the rainbow disconnection number of $G$, denoted by rd$(G)$, is defined as the smallest number of colors that are needed in order to make $G$ rainbow disconnected. 
In this paper, we first give some tight upper bounds for rd$(G)$, and moreover, we completely characterize the graphs which meet the upper bound of the Nordhaus-Gaddum type results obtained early by us. Secondly, we propose a conjecture that $\lambda^+(G)\leq \textnormal{rd}(G)\leq \lambda^+(G)+1$, where $\lambda^+(G)$ is the upper edge-connectivity, and prove the conjecture for many classes of graphs, to support it. Finally, we give the relationship between rd$(G)$ of a graph $G$ and the rainbow vertex-disconnection number rvd$(L(G))$ of the line graph $L(G)$ of $G$.

Bounds for the rainbow disconnection number of graphs

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set of G is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S such that when x and y are nonadjacent, S is rainbow and x and y belong to different components of $$G-S$$
 ; whereas when x and y are adjacent, $$S+x$$
 or $$S+y$$
 is rainbow and x and y belong to different components of $$(G-xy)-S$$
 . Such a vertex subset S is called an x–y rainbow vertex-cut of G. For a connected graph G, the rainbow vertex-disconnection number of G, denoted by rvd(G), is the minimum number of colors that are needed to make G rainbow vertex-disconnected. In this paper, we obtain bounds of the rainbow vertex-disconnection number of a graph in terms of the minimum degree and maximum degree of the graph. We give a tighter upper bound for the maximum size of a graph G with $$rvd(G)=k$$
 for $$k\ge \frac{n}{2}$$
 . We then characterize the graphs of order n with rainbow vertex-disconnection number $$n-1$$
 and obtain the maximum size of a graph G with $$rvd(G)=n-1$$
 . Moreover, we get a sharp threshold function for the property $$rvd(G(n,p))=n$$
 and prove that almost all graphs G have $$rvd(G)=rvd({\overline{G}})=n$$
 . Finally, we obtain some Nordhaus–Gaddum-type results: $$n-5\le rvd(G)+rvd({\overline{G}})\le 2n$$
 and $$n-1\le rvd(G)\cdot rvd({\overline{G}})\le n^2$$
 for the rainbow vertex-disconnection numbers of nontrivial connected graphs G and $${\overline{G}}$$
 with order $$n\ge 24$$
 .

Further Results on the Rainbow Vertex-Disconnection of Graphs

Foucaud et al. recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. For every edge e of G and a set M⊆V(G), M is a distance-edge-monitoring (DEM for short) set if there are a vertex x of M and a vertex y of G such that e belongs to all shortest paths between x and y. The DEM number dem(G) is the smallest size of such a set in G. The vertices of M represent distance probes in a network modeled by G; when the edge e fails, the distance from x to y increases, and thus we are able to detect the failure. In this paper, we study Erdös–Gallai-type problems for DEM numbers of general graphs. The exact values or bounds of dem(G) for radix n-triangular mesh networks and hexagonal networks are also given. 

Erdös–Gallai-type problems for distance-edge-monitoring numbers

For an edge-colored graph G, a set F of edges of G is called a proper edge-cut if F is an edge-cut of G and any pair of adjacent edges in F are assigned different colors. An edge-colored graph is proper disconnected if for each pair of distinct vertices of G there exists a proper edge-cut separating them. For a connected graph G, the proper disconnection number of G, denoted by pd(G), is the minimum number of colors that are needed in order to make G proper disconnected. In this paper, we first give the exact values of the proper disconnection numbers for some special families of graphs. Next, we obtain a sharp upper bound of pd(G) for a connected graph G of order n, i.e, $$pd(G)\le \min \{ \chi '(G)-1, \left\lceil \frac{n}{2} \right\rceil \}$$
 . Finally, we show that for given integers k and n, the minimum size of a connected graph G of order n with $$pd(G)=k$$
 is $$n-1$$
 for $$k=1$$
 and $$n+2k-4$$
 for $$2\le k\le \lceil \frac{n}{2}\rceil $$
 .

Proper Disconnection of Graphs

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set of G is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S of G such that when x and y are nonadjacent, S is rainbow and x and y belong to different components of G − S; whereas when x and y are adjacent, S + x or S + y is rainbow and x and y belong to different components of (G − xy) − S. For a connected graph G, the rainbow vertex-disconnection number of G, denoted by rvd(G), is the minimum number of colors that are needed to make G rainbow vertex-disconnected. In this paper, we characterize all graphs of order n with rainbow vertex-disconnection number k for k ∈ {1, 2, n}, and determine the rainbow vertex-disconnection numbers of some special graphs. Moreover, we study the extremal problems on the number of edges of a connected graph G with order n and rvd(G)= k for given integers k and n with 1 ≤ k ≤ n.

The Rainbow Vertex-disconnection in Graphs

For an edge-colored graph G, a set F of edges of G is called a proper edge-cut if F is an edge-cut of G and any pair of adjacent edges in F are assigned by different colors. An edge-colored graph is called proper disconnected if for each pair of distinct vertices of G there exists a proper edge-cut separating them. For a connected graph G, the proper disconnection number of G, denoted by pd(G), is defined as the minimum number of colors that are needed to make G proper disconnected. In this paper, we first show that it is NP-complete to decide whether a given k-edge-colored graph G with $\varDelta (G)=4$ is proper disconnected. Then, for a graph G with $\varDelta (G)\le 3$ we show that $pd(G)\le 2$ and determine the graphs with $pd(G)=1$ and 2 in polynomial time, respectively, when the set of vertices with degree 3 in G is an independent set. Finally, we show that for a general graph G, deciding whether $pd(G)=1$ is NP-complete, even if G is bipartite.

Complexity Results for the Proper Disconnection of Graphs

Complexity results for the proper disconnection of graphs

Let $G$ be a nontrivial connected and vertex-colored graph. A subset $X$ of the vertex set of $G$ is called rainbow if any two vertices in $X$ have distinct colors. The graph $G$ is called \emph{rainbow vertex-disconnected} if for any two vertices $x$ and $y$ of $G$, there exists a vertex subset $S$ of $G$ such that when $x$ and $y$ are nonadjacent, $S$ is rainbow and $x$ and $y$ belong to different components of $G-S$; whereas when $x$ and $y$ are adjacent, $S+x$ or $S+y$ is rainbow and $x$ and $y$ belong to different components of $(G-xy)-S$. For a connected graph $G$, the \emph{rainbow vertex-disconnection number} of $G$, denoted by $rvd(G)$, is the minimum number of colors that are needed to make $G$ rainbow vertex-disconnected. 
In this paper, we characterize all graphs of order $n$ with rainbow vertex-disconnection number $k$ for $k\in\{1,2,n\}$, and determine the rainbow vertex-disconnection numbers of some special graphs. Moreover, we study the extremal problems on the number of edges of a connected graph $G$ with order $n$ and $rvd(G)=k$ for given integers $k$ and $n$ with $1\leq k\leq n$.

The rainbow vertex-disconnection in graphs

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.

Monochromatic disconnection: Erdős-Gallai-type problems and product graphs

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