MONOTONE CHROMATIC NUMBER OF GRAPHS

Abstract. For a graph G = (V,E), a vertex coloring (or, simply, a coloring) of G is a function C : V (G) → {1, 2, ..., k} (using the non-negative integers {1, 2, ..., k} as colors). In this research work, we introduce a new type of graph coloring called monotone coloring, along with this new coloring, we define the monotone chromatic number of a graph and establish some related new graphs. Basic properties and exact values of the monotone chromatic number of some graph families, like standard graphs, Kragujevac trees and firefly graph are obtained. Also, we get a characterization for bipartite graphs by defining the monotone bipartite graph. Exact values of the monotone chromatic number for some special case of Cartesian product of graphs are found. Finally, upper and lower bounds for monotone chromatic number of the Cartesian product for non trivial connected graphs are presented.


Introduction
Throughout this research work, by graph we mean finite graph without loops and parallel edges. Any notations or terminology not specifically defined here, we refer the book [6]. More details about coloring in graph and its related are reported in ( [3,8]). Two interesting types of coloring are introduced and studied in ( [1,2,5]). As usual, P n , C n , K n and W n are the n−vertex path, cycle, complete, and wheel graph, respectively, K r,s is the complete bipartite graph on r + s vertices and S r is the star graph with r + 1 vertices.
Graph coloring is one of essential concepts in the theory of graphs. It has preoccupied a large number of people as a distraction puzzle during the 19th century and later in the framework of scientific research, since this conception exhibits a significant interest from a theoretical and practical point of view. Many applications are modeled and investigated with the use of graph coloring.
The huge applications of coloring motivated us to introduce a new type of graph coloring called monotone coloring of the graph, we define the monotone chromatic number of a graph, complete monotone graph and monotone clique set of a graph. Some basic properties and relations with the other graph parameters and exact values of the monotone chromatic number of some graph families, like standard graphs, firefly graph and Kragujevac trees obtained. Also, we get a characterization for bipartite graphs by defining the monotone bipartite graph. Exact values of the monotone chromatic number for some special case of Cartesian product of graphs are found. Lastly upper and lower bounds for monotone chromatic number of the Cartesian product for non trivial connected graphs are presented.

Monotone Chromatic Number of Graphs
In this section, we define the monotone chromatic coloring and monotone chromatic number of a graph and give several preliminary results and straightforward facts regarding the monotone coloring of graphs.
Also we found the monotone chromatic number for some families of graphs.
) for all i = 1, 2, ..., k. Any two vertices u and v in G are called monotone adjacent if there exists a monotone path connected them.
Definition 2.2. A monotone k− coloring of the graph G is coloring the vertices of G with k colors such that no two monotone adjacent vertices share the same color. The smallest integer k such that G has a monotone k− kcoloring is called the monotone chromatic number of G and denoted by χ mo (G). A graph G is said to be monotone k− colorable if it has monotone k− coloring.
Monotone coloring as function we can define as:  (1) For any path P n with n ≥ 2 vertices, (2) For any connected regular graph G with n ≥ 3 vertices, χ mo (G) = n.
(3) For any complete bipartite graph K r,s , where 1 < r ≤ s, we have χ mo K r,s = 2.
(7) For any gear graph G with 2n + 1 vertices, we have From the definition of the proper coloring and monotone coloring, it is obviously, that any monotone coloring is proper coloring but the converse is not true. As any two adjacent vertices in any graph are also monotone adjacent, then we have the following result. (1) For any monotone χ mo (G)− coloring of any graph G, all the very weak vertices has the same color.
Remark 2.1. Let G and H be any two graphs such that H is subgraph of G. Then the monotone chromatic number of G and H are not comparable. That means all the possibilities allowed.
According to the monotone adjacency between vertices, we define the monotone bipartite graph, and complete monotone graph. The definition of monotone bipartite graph characterize the trees into two families monotone trees and non-monotone trees The double star graph denoted by B(r, s) with r +s+ 2 vertices, is a tree that containing exactly two non-pendent vertices. Proof. Let G be a graph such that χ mo (G) = 2 and suppose contrary that there is monotone path uvw of length 2, we need at least three colors for monotone coloring of G that means χ mo (G) ≥ 3 which is a contradiction. Therefore any monotone is of length at most one. Hence G is monotone bipartite graph. To prove the other direction, suppose, that G is monotone bipartite graph. Then the result coming from the Proposition 2.2. Proof. Suppose that χ mo (G) = 2, and assume in contrary that, there is a vertex v ∈ V (G) which is neither very strong nor very weak. Then we have two cases: then v is very weak which contradict our assumption.  To prove the other direction. Suppose G contains at most either one very weak or one very strong vertex.
then clearly there exist a monotone path containing all the vertices that means between any two vertices there is a monotone. Hence G is monotone complete graph. construct the rooted tree B k by identifying the roots of k copies of P 3 . The vertex obtained by identifying the roots of P 3 −trees is the root of B k .   Case 2. The central vertex v is very weak, then we can define monotone coloring function C mo by partition the vertex set into 4 subsets, S 1 is the pendant vertices, S 2 the support vertices and S 3 be the roots vertices of the branches and S 4 be the central vertex v and giving color for each subset as following: Therefore, χ mo (G) ≤ 3 and since the set of vertices in any path between the pendant vertex of any branch and its root vertex is monotone clique set. Hence in this case χ mo (G) = 3. and one color to all the vertices of S 2 and another different color to the vertices of S 1 , so, we need to d + 3 colors to this monotone coloring function, therefore χ mo (G) ≤ d + 3. Obviously, if we take any pendant vertex along with its support vertex in any branches along with the set S 3 will make monotone clique set of Also, it is obviously to see that the vertices of S 3 with one pendent vertex with its support vertex and one root vertex from S 4 will make clique set in G. Hence χ mo (G) = t + 4.  Proof. Let G be any nontrivial connected graph with monotone chromatic number χ mo (G) = s. Suppose to the contrary, that we have maximum clique set A with size |A| = ω mo (G) = t, where either t < s or t > s. Case 1. If t < s. Then there exists at least one coloring class say B and at least one monotone path between every two elements one from A and one from B which is contradict that A is the monotone clique with maximum size.
Case 2. If t > s, then at least we need to t + 1 colors for monotone coloring which is contradict that Theorem 2.4. Let G ∼ = K m1,m2···m k , where m 1 ≤ m 2 ≤ · · · ≤ m k be any complete k− partite graph and there are t i partite sets of the same number of vertices λ i , where i = 1, 2, · · · , s for some positive integer i. Then Proof. Let G ∼ = K m1,m2···m k , where m 1 ≤ m 2 ≤ · · · ≤ m k be any complete k− partite graph and there are t i partite sets of the same number of vertices λ i , where i = 1, 2, · · · , s for some positive integer i ,by reordering the partite sets which they have different number of elements as we can define a monotone coloring function as the following.
by assigning the color i for each vertex in V i and assigning for each vertex in the equal parite sets to different colors that means assign Clearly, the function f which defined above is monotone coloring function on G. Therefore, It is not difficult to check that the set which contains the vertices in and only one vertex from each V i will make clique set with k + Hence by inequalities 2.1 and 2.2, we get, Theorem 2.5. Let G be any connected graph with n ≥ 3 vertices and D is the set of distinct degrees of the vertices. Then G is monotone complete graph if and only if for any set of vertices with the same degree in G is clique set in G.
Proof. Let G be monotone complete graph with n ≥ 3 vertices, since G is monotone complete, then ω mo (G) = n. Therefore V (G) is a monotone clique set contains all the vertices of G.
If G is regular, then there is only one clique set containing all the vertices. Suppose the graph is not regular that means the set D is of size greater than or equal to 2. Let D = {k 1 , k 2 , . . . , k t } for some integer t ≥ 2, also let C 1 , C 2 , . . . , C t be the sets of vertices of degrees k 1 , k 2 , . . . , k t respectively. Suppose in contrary, that there exists one set C j , 1 ≤ j ≤ t such that C j not a monotone clique set, then there exist at least two vertices u and v which they are not monotone adjacent in G which is a contradict that V (G) is clique set.
Similarly, let C 1 , C 2 , . . . , C t are monotone clique sets in G. suppose G is not monotone complete, then there exist at least two vertices u and v not monotone adjacent if they are with the same degree, then they will belong to same set C j , 1 ≤ j ≤ t which is a contradiction, similarly we will get a contradiction even if they have different degrees.
We recall, that in [7], a firefly graph F s,t,l where s ≥ 0, t ≥ 0, n − 2s − 2t − 1 ≥ 0 is a graph of order n that consists of s triangles, t pendent paths of length 2 and l pendant edges sharing a common vertex (see Figure 1).
3, x is the central vertex.
Therefore, χ mo (G) ≤ 3. The set of vertices in any triangle in the firefly graph generate a monotone clique set, that means χ mo (G) ≥ 3. Hence, χ mo (G) = 3.
The corona product G 1 • G 2 of two graphs G 1 and G 2 , where V (G 1 ), V (G 2 ) are the set of vertices of G 1 , G 2 respectively, is the graph obtained by taking |V (G 1 )| copies of G 2 and joining each vertex of the i-th copy with the corresponding vertex u ∈ V (G 1 ) [6].
Proposition 2.8. For any positive integer a ≥ 3, there exists a graph G such that χ(G) = a and χ mo (G) = a + 1.
Proof. Let G be a graph which construct by the corona product between the complete graphs K a and K 1 .
Then it is not difficult to see that χ(G) = a and χ mo (G) = a + 1.
Definition 2.8. A subset S ⊆ V is called monotone independent set, if the set S does not contains any monotone adjacent vertices. The maximum cardinality of the monotone independent set is called monotone independence number of the graph and denoted by β mo (G). For example; for any path P n with n ≥ 3 vertices, β mo (P n ) = 2. The monotone independence number of any complete monotone graph is one.
Theorem 2.6. Let G be a graph with n vertices and with monotone independence number β mo (G). Then Further, the equality holds if G is complete monotone graph.

Monotone Chromatic number for Some Cartesian Product of Graphs
The Cartesian product G 1 G 2 of two graphs G 1 and G 2 , where V (G 1 ), E(G 1 ) and V (G 2 ), E(G 2 ) are the sets of vertices and edges of G 1 and G 2 , respectively, has the vertex set V (G 1 ) × V (G 2 ) and two vertices (u, u ) and (v, v ) are connected by an edge if and only if either (u = v and u v ∈ E(G 2 )) or (u = v and uv ∈ E(G 1 )) [6].
We can define a monotone coloring function as following Clearly f is a monotone-k-coloring function, where k = (r − 2)(s − 1) + 1. So The set S 2 ∪ {v 1,1 } is monotone clique set with (r − 2)(s − 1) + 1 vertices. Therefore, Hence by inequalities 3.1 and 3.2, we get, Proof. Let G ∼ = C r P s , where r, s ≥ 3. By labeling the vertices of the internal and external cycles of G as the vertices of the internal cycle by v 1 , v 2 , . . . , v r and the vertices of external cycle by u 1 , u 2 , . . . , u r see as Figure 2. Let us partition the set of vertices as following subsets: By defining a the coloring function, f : V (G) −→ {1, 2, . . . , r(s − 1)} such that: for any vertex x Obviously, f is a monotone r(s − 1) coloring function. So The set S 1 ∪ S 3 is monotone clique set with r(s − 1) vertices. Therefore, Hence by inequalities 3.3 and 3.4, we get, The stacked book is defined as the graph which construct by Cartesian product between star S r of r + 1 vertices and path P s of s vertices and denoted by B r,s . That means B r,s ∼ = S r P s .  Let us partition the vertex set of the graph G to following subsets: j < s}, S 3 = {v 1j ; 2 < j < s}, S 4 = {v i2 ; 1 ≤ i ≤ r}, S 5 = {v ij ; 2 ≤ i ≤ r, and 2 < j < s} and S 6 = {v 0s }.
(2) x and u are monotone adjacent vertices in G and y = v in H.
(3) x is monotone adjacent with u in G and y is monotone adjacent with v in H.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.