A Survey on Monochromatic Connections of Graphs

The concept of monochromatic connection of graphs was introduced by Caro and Yuster in 2011. Recently, a lot of results have been published about it. In this survey, we attempt to bring together all the results that dealt with it. We begin with an introduction, and then classify the results into the following categories: monochromatic connection coloring of edge-version, monochromatic connection coloring of vertex-version, monochromatic index, monochromatic connection coloring of total-version.


Introduction
All graphs considered in this paper are simple, finite and undirected. We follow the terminology and notation of Bondy and Murty in [3]. Let H be a nontrivial connected  [7,4,20,21,22], where in the latter we seek to find an edge-coloring with minimum number of colors so that there is a rainbow path joining any two vertices; see [21,22] for details. As introduced by Caro and Yuster [9], for a connected graph G, the monochromatic connection number of G, denoted by mc(G), is the maximum number of colors that are needed in order to make G monochromatically connected. An extremal MC-coloring is an MC-coloring that uses mc(G) colors.
Gonzlez-Moreno, Guevara, and Montellano-Ballesteros [12] generalized the above concept to digraphs. Let H be a nontrivial strongly connected digraph with an arc- of H is called a k-monochromatic connection coloring (MX k -coloring) if it makes H kmonochromatically connected. For a connected graph G and a given integer k such that 2 ≤ k ≤ |V (G)|, the k-monochromatic index mx k (G) of G is the maximum number of colors that are needed in order to make G k-monochromatically connected. An extremal MX k -coloring is an MX k -coloring that uses mx k (G) colors. By definition, Note that the above graph-parameters are defined on edge-colored graphs. Naturally, Cai it makes H vertex-monochromatically connected. For a connected graph G, the vertexmonochromatic connection number of G, denoted by vmc(G), is the maximum number of colors that are needed in order to make G vertex-monochromatically connected. An extremal VMC-coloring is a VMC-coloring that uses vmc(G) colors.
Li and Wu [23] introduced another graph-parameter corresponding to the k-monochromatic index, which is defined on vertex-colored graphs. A tree T in a vertexcolored graph H is called a vertex-monochromatic tree if all the internal vertices of T have the same color. For an S ⊆ V (H), a vertex-monochromatic S-tree is a vertex-monochromatic tree of H containing the vertices of S. Given an integer k with 2 ≤ k ≤ |V (H)|, the graph H is called k-vertex-monochromatically connected if for any set S of k vertices of H, there exists a vertex-monochromatic Stree in H. A vertex-coloring of H is called a k-vertex-monochromatic connection coloring (V MX k -coloring) if it makes H k-vertex-monochromatically connected. For a connected graph G and a given integer k such that 2 ≤ k ≤ |V (G)|, the kvertex-monochromatic index vmx k (G) of G is the maximum number of colors that are needed in order to make G k-vertex-monochromatically connected. An extremal V MX k -coloring is a V MX k -coloring that uses vmx k (G) colors. By definition, we have Jiang, Li and Zhang [15] introduced the monochromatic connection of total-coloring version. Let H be a nontrivial connected graph with a total-coloring f : V (H) ∪ E(H) → {1, 2, . . . , ℓ} (ℓ is a positive integer), where any two elements may be colored the same. A path in a total-colored graph H is a total-monochromatic path if all the edges and internal vertices of the path are colored with a same color. The graph H is called total-monochromatically connected if any two vertices of H are connected by a total-monochromatic path. A total-coloring of H is a total-monochromatic connection coloring (TMC-coloring) if it makes H total-monochromatically connected. For a connected graph G, the total-monochromatic connection number of G, denoted by tmc(G), is the maximum number of colors that are needed in order to make G totalmonochromatically connected. An extremal TMC-coloring is a TMC-coloring that uses tmc(G) colors.
Next, we recall the definitions of various products of graphs, which will be used in the sequel. The Cartesian product of two graphs G and H, denoted by G✷H, is defined to have the vertex-set V (G)×V (H), in which two vertices (g, h) and (g ′ , h ′ ) are adjacent if and only if g = g ′ and hh ′ ∈ E(H), or h = h ′ and gg ′ ∈ E(G). The lexicographic product G • H of two graphs G and H has the vertex-set and two vertices (g, h), (g ′ , h ′ ) are adjacent if and only if gg ′ ∈ E(G), or g = g ′ and hh ′ ∈ E(H). The strong product G ⊠ H of two graphs G and H has the vertex-set V (G) × V (H). Two vertices (g, h) and (g ′ , h ′ ) are adjacent whenever gg ′ ∈ E(G) and h = h ′ , or g = g ′ and hh ′ ∈ E(H), or gg ′ ∈ E(G) and hh ′ ∈ E(H). The direct product G ×H of two graphs G and H has the vertex-set V (G) ×V (H). Two vertices (g, h) and (g ′ , h ′ ) are adjacent if the projections on both coordinates are adjacent, i.e., gg ′ ∈ E(G) and hh ′ ∈ E(H). Finally, the join G + H of two graphs G and H has the vertex-set The most frequently occurring probability models of random graphs is the Erdös-Rényi random graph model G(n, p) [10]. The model G(n, p) consists of all graphs with n vertices in which the edges are chosen independently and with probability p. We say an event A happens with high probability if the probability that it happens approaches 1 as n → ∞, i.e., P r[A] = 1 − o n (1). Sometimes, we say w.h.p. for short. We will always assume that n is the variable that tends to infinity. Let G and H be two graphs on n vertices. A property P is said to be monotone if whenever G ⊆ H and G satisfies P , then H also satisfies P . For a graph property P , a function p(n) is called a threshold function of P if: • for every r(n) = ω(p(n)), G(n, r(n)) w.h.p. satisfies P ; and • for every r ′ (n) = o(p(n)), G(n, r ′ (n)) w.h.p. does not satisfy P . Furthermore, p(n) is called a sharp threshold function of P if there exist two positive constants c and C such that: • for every r(n) ≥ C · p(n), G(n, r(n)) w.h.p. satisfies P ; and • for every r ′ (n) ≤ c · p(n), G(n, r ′ (n)) w.h.p. does not satisfy P .
It is well known that all monotone graph properties have a sharp threshold function; see [2] and [11] for details. (a) G is 4-connected.
(e) G has a cut vertex.
Jin, Li and Wang got some conditions on graphs containing triangles. Caro and Yuster [9] also showed some nontrivial upper bounds for mc(G) in terms of the chromatic number, the connectivity, and the minimum degree. Recall that a graph is called s-perfectly-connected if it can be partitioned into s + 1 parts {v}, V 1 , . . . , V s , such that each V j induces a connected subgraph, any pair V j , V r induces a corresponding complete bipartite graph, and v has precisely one neighbor in each V j . Notice that such a graph has minimum degree s, and v has degree s.
(3) If G is not k-connected, then mc(G) ≤ m − n + k. This is sharp for any k.
As an application of Theorem2.5(4), Caro and Yuster got the upper bounds for the following planar graphs.

Erdős-Gallai-type problems for mc(G)
Cai, Li and Wu [5] studied the following two kinds of Erdős-Gallai-type problems for mc(G).
Problem A. Given two positive integers n and k with 1 ≤ k ≤ n 2 , compute the minimum integer f (n, k) such that for any graph G of order n, if |E(G)| ≥ f (n, k) then mc(G) ≥ k.
Problem B. Given two positive integers n and k with 1 ≤ k ≤ n 2 , compute the maximum integer g(n, k) such that for any graph G of order n, if |E(G)| ≤ g(n, k) It is worth mentioning that the two parameters f (n, k) and g(n, k) are equivalent to another two parameters. Let t(n, k) = min{|E(G)| : |V (G)| = n, mc(G) ≥ k} and s(n, k) = max{|E(G)| : |V (G)| = n, mc(G) ≤ k}. It is easy to see that t(n, k) = g(n, k − 1) + 1 and s(n, k) = f (n, k + 1) − 1. In [5] the authors determined the exact values of f (n, k) and g(n, k) for all integers n, k with 1 ≤ k ≤ n 2 .

Results for graph classes
From the above theorems, they also verified the following corollary.
Corollary 2.16. [14] Let G be a connected graph of order n. Then

Results for graph products
Mao, Wang, Yanling and Ye [24] studied the monochromatic connection numbers of the following graph products.   As an application of the above results, they also studied the following graph classes.
We call P n ✷P m a two-dimensional grid graph, where P n and P m are paths on n and m vertices, respectively.
An n-dimensional mesh is the Cartesian product of n linear arrays. Particularly, two-dimensional grid graph is a 2-dimensional mesh. An n-dimensional hypercube is an n-dimensional mesh, in which all the n linear arrays are of size 2.
(2) For network P L 1 • P L 2 • · · · • P Ln , An n-dimensional torus is the Cartesian product of n cycles R 1 , R 2 , · · · , R n of size at least three.
where r i is the order of R i and 3 ≤ i ≤ n.
(2) For network Let K m be a clique of m vertices, m ≥ 2. An n-dimensional generalized hypercube is the Cartesian product of n cliques.
(2) For network K m 1 • K m 2 • · · · • K mn , An n-dimensional hyper Petersen network HP n is the Cartesian product of Q n−3 and the well-known Petersen graph , where n ≥ 3 and Q n−3 denotes an (n − 3)dimensional hypercube.
The network HL n is the lexicographical product of Q n−3 and the Petersen graph, where n ≥ 3 and Q n−3 denotes an (n − 3)-dimensional hypercube.
For the join of two graphs, Jin, Li and Wang got the following results.
Theorem 2.31. [18] Let G be the join of a connected graph G 1 and a disconnected Theorem 2.32. [18] Let G be the join of two disjoint disconnected graphs G 1 and G 2 . Then

Results for random graphs
The goal of MC-coloring of a graph is to find as many as colors to make the graph monochromatically connected. So it is interesting to consider the threshold function of property mc (G (n, p)) ≥ f (n), where f (n) is a function of n. For any graph G with n vertices and any function f (n), having mc(G) ≥ f (n) is a monotone graph property (adding edges does not destroy this property), so it has a sharp threshold function.
Gu, Li, Qin and Zhao [14] showed a sharp threshold function for mc(G) as follows.

Upper and lower bounds for vmc(G)
For a connected graph G of order 1 or 2, it is easy to check vmc(G) = 1, 2, respectively.
For a connected graph G of order at least 3, Cai, Li and Wu [6] got that a general lower bound for vmc(G) is ℓ(T ) + 1 ≥ 3, where T is a spanning tree of G, and ℓ(T ) is the number of leaves in T . Simply take a spanning tree T of G. Then, give all the non-leaves in T one color, and each leaf in T a distinct new color. Clearly, this is a VMC-coloring of G using ℓ(T ) + 1 colors.
They also got an upper bound for vmc(G).  (2) If d ≥ 3, then vmc(G) ≤ n − d + 2 , and the bound is sharp.
Problem A: Given two positive integers n, k with 3 ≤ k ≤ n, compute the minimum integer f v (n, k) such that for any graph G of order n, if |E(G)| ≥ f v (n, k) then vmc(G) ≥ k.
Problem B: Given two positive integers n, k with 3 ≤ k ≤ n, compute the maximum integer g v (n, k) such that for any graph G of order n, if |E(G)| ≤ g v (n, k) then vmc(G) ≤ k.
Note that g v (n, n) = n 2 , and g v (n, k) does not exist for 3 ≤ k ≤ n − 1. This is because for a star S n on n vertices, we have vmc(S n ) = n. For this reason, Cai, Li and Wu [6] just studied Problem A. They got the value of f v (n, k).
Moreover, these bounds are sharp.

The arc-coloring version for digraphs
Gonzlez-Moreno, Guevara, and Montellano-Ballesteros [12] got the following result for strongly connected oriented graph. As an application of Theorem 4.1, they found a sufficient and necessary condition to determine whether a strongly connected oriented graph is Hamiltonian.

Vertex version
Li and Wu [23] studied the hardness for computing vmx k (G). They showed that given a connected graph G = (V, E), and a positive integer L with L ≤ |V |, to decide whether vmx k (G) ≥ L is NP-complete for each k with 2 ≤ k ≤ |V |. In particular, computing vmx k (G) is NP-hard.

Nordhaus-Gaddum-type results
Recall that Cai, Li and Wu [6] got the Nordhaus-Gaddum-type result for vmc(G). Li and Wu [23] got the following Nordhaus-Gaddum-type lower bounds of vmx k for k with 3 ≤ k ≤ n.
Theorem 5.2. [23] Suppose that both G and G are connected graphs on n vertices.
They also got the following Nordhaus-Gaddum-type upper bound of vmx k for k with ⌈ n 2 ⌉ ≤ k ≤ n.
Theorem 5.3. [23] Suppose that both G and G are connected graphs on n ≥ 5 vertices.
Then, for any k with ⌈ n 2 ⌉ ≤ k ≤ n, we have that vmx k (G) + vmx k (G) ≤ 2n − 2, and this bound is sharp. 6 The total-coloring version Jiang, Li and Zhang [16] studied the hardness for computing tmc(G). They showed that given a connected graph G = (V, E), and a positive integer L with L ≤ |V | + |E|, to decide whether tmc(G) ≥ L is NP-complete. In particular, computing tmc(G) is NP-hard.

Upper and low bounds for tmc(G)
Let l(T ) denote the number of leaves in a tree T . For a connected graph G, let l(G) = max{ l(T ) | T is a spanning tree of G }. Jiang, Li and Zhang [15] got the following lower bound of tmc(G). They also gave some sufficient conditions for graphs attaining this lower bound.  Jiang, Li and Zhang [15] computed the total monochromatic connection numbers of wheel graphs and complete multipartite graphs. Note that tmc(C 5 ) = 4 < vmc(C 5 ) = 5, where m < 2n − d − 2 and ∆ < n+1 2 . This implies that the conditions of Theorems 6.5 and 6.6 cannot be improved. If G is a star, then tmc(G) = vmc(G) = n. However, they could not show whether there exist other graphs with tmc(G) ≤ vmc(G). Then they proposed the following problem. Problem 6.7. [15] Dose there exists a graph of order n ≥ 6 except for the star graph such that tmc(G) ≤ vmc(G)?
In addition, they proposed the following conjecture.   Figure 3.

Results for random graphs
For a property P of graphs and a positive integer n, define P rob(P, n) to be the ratio of the number of graphs with n labeled vertices having property P over the total number of graphs with these vertices. If P rob(P, n) approaches 1 as n tends to infinity, then we say that almost all graphs have property P . More details can be found in [1]. Jiang, Li and Zhang [15] got the following result for tmc(G).
Theorem 6.17. [15] For almost all graphs G of order n and size m, we have tmc(G) = m − n + 2 + l(G).
Jiang, Li and Zhang [16] showed a sharp threshold function for tmc(G) as follows. (1) is a sharp threshold function for the property tmc(G(n, p)) ≥ f (n).
Remark 6.19. Note that if f (n) = 1 2 n(n − 1) + n, then G(n, p) is a complete graph K n and p = 1. Hence we only concentrate on the case f (n) < 1 2 n(n − 1) + n.

Erdős-Gallai-type problems for tmc(G)
Jiang, Li and Zhang [17] studied the following two kinds of Erdős-Gallai-type problems for tmc(G).
Problem A. Given two positive integers n and k with 3 ≤ k ≤ n 2 + n, compute the minimum integer f T (n, k) such that for any graph G of order n, if |E(G)| ≥ f T (n, k) then tmc(G) ≥ k.
Problem B. Given two positive integers n and k with 3 ≤ k ≤ n 2 + n, compute the maximum integer g T (n, k) such that for any graph G of order n, if |E(G)| ≤ g T (n, k) then tmc(G) ≤ k.
They completely determined the values of f T (n, k) and g T (n, k).

Concluding remarks
It is easily seen also from Theorem 2.1 (a) that for almost all connected graphs G it holds that mc(G) = m(G) − n(G) + 2.
Another problem is to consider more monochromatic paths connecting a pair of vertices. The definitions can be easily given as follows. An edge-colored graph is called monochromatically k-connected if each pair of vertices of the graph is connected by k monochromatic paths in the graph. For a k-connected graph G, the monochromatic k-connection number, denoted by mc k (G), is defined as the maximum number of colors that are needed in order to make G monochromatically k-connected. As far as we knew, there is no paper published on this parameter. We think that to get some bounds for the case k = 2 is already quite interesting and not so easy.
It is seen that results for the monochromatic indices are very few, and more efforts are needed for deepening the research. It is also seen that research on smc(D) for digraphs has just started, and one can develop it with many possibilities.
Finally, we point out that we changed some terminology and notation. For examples, we use vmc(G) to replace mvc(G) and vmx k (G) to replace mvx k (G), etc.
This is because we think that the term "vertex-monochromatic connection" is better than "monochromatic vertex-connection". This is just a matter of taste, depending on authors and readers.