On the r-dynamic coloring of the direct product of a path with either a complete graph or a wheel graph

In this paper, it is explicitly determined the r-dynamic chromatic number of the direct product of any given path with either a complete graph or a wheel graph. Illustrative examples are shown for each one of the cases that are studied throughout the paper.


Introduction
In 2001, Bruce Montgomery [1] (see also [2]) introduced the concept of r-dynamic proper kcoloring of a graph G = (V(G), E(G)) as any proper k-coloring c : V(G) → {0, . . . , k − 1} of such a graph such that |c(N(v))| ≥ min{r, d(v)}, (1.1) for all v ∈ V(G). Here, N(v) and d(v) denote, respectively, the neighborhood and the degree of the vertex v. In addition, he introduced the notion of r-dynamic chromatic number χ r (G) of the graph G as the minimum positive integer k for which an r-dynamic proper k-coloring of G exists. As such, these concepts constitute a natural generalization of the classical notions of proper coloring and the chromatic number χ(G) of a graph G, which arise when r = 1. Montgomery himself proposed the natural question about how much the difference between χ r (G) and χ(G) varies, for every r > 1, and also whether such a difference is bounded for all graphs. Concerning this last question, he proved [1] the existence of graphs for which this difference is unbounded even for r = 2. Further, Montgomery also introduced the study of the r-dynamic chromatic number of specific families of graphs, for all r > 1. More specifically, he determined explicitly all these values for any complete graph, cycle or tree [1] and dealt with the case r = 2 for any multipartite graph [2].
Since the original manuscript of Montgomery, a wide amount of graph theorists have dealt with the study of upper bounds concerning the r-dynamic chromatic number of any graph. In this regard, Montgomery himself [1] proved that χ 2 (G) ≤ ∆(G) + 3, for every graph G, where ∆(G) denotes the maximum vertex degree in G. Shortly after, Lai et al. [2] proved that χ 2 (G) ≤ ∆(G) + 1, if ∆(G) ≥ 4. In addition, Montgomery [1] also conjectured that χ 2 (G) ≤ χ(G) + 2, for every regular graph G. This inequality was proved by Lai et al. [3] for graphs that are connected and claw-free. Concerning the conjecture itself, it was proved by Akabari et al. [4] in case of dealing with bipartite regular graphs. Furthermore, Alishahi [5] proved the existence of a constant c such that χ 2 (G) ≤ χ(G) + c · ln(k) + 1, for every k-regular graph G. This last author [6] also proved that χ 2 (G) ≤ χ(G) + γ(G), where γ(G) denotes the domination number of G. For a general positive integer r, Jahanbekam et al. [7] proved that χ r (G) ≤ r · ∆(G) + 1. For r ≥ 2, Lai et al. [3] had already proved that χ r (G) ≤ ∆(G) + r 2 − r + 1, whenever ∆(G) ≤ r. Finally, concerning the study of upper bounds of the r-dynamic chromatic number of products of graphs, Akbari et al [8] proved that χ 2 (G H) ≤ max{χ 2 (G), χ 2 (H)}, if δ(G) ≥ 2, where G H denotes the Cartesian product of two graphs G and H, and δ(G) denotes the minimum vertex degree of the graph G.
In the recent literature, it is also remarkable the acquired relevance of determining explicitly the r-dynamic chromatic number of specific families of graphs, for every positive integer r. It is so that this value has already been studied for grid graphs [7,9]; helm graphs [10]; prism graphs, threecyclical ladder graphs, joint graphs and circulant graphs [11]; toroidal graphs [12]; some cycle-related graphs [13]; coronations of paths [14]; and subdivision-edge coronas of a path [15]. In addition, it has also been studied the r-dynamic chromatic number of the Cartesian product and the corona product of distinct types of graphs [16][17][18][19][20][21][22]. Nevertheless, the r-dynamic chromatic number of direct products of graphs has only recently been given attention. More specifically, it has explicitly been determined [23] for the direct product of any given path with either a path or a cycle. This paper delves into this topic by determining the r-dynamic chromatic number of the direct product of any given path with either a complete graph or a wheel graph.
The paper is organized as follows. In Section 2, we describe some preliminary concepts and results on Graph Theory that are used throughout the paper. Then, Sections 3 and 4 deal, respectively, with the r-dynamic chromatic number of the direct product of a path with either a complete graph or a wheel graph.

Preliminaries
This section deals with some preliminary concepts and results on Graph Theory that are used throughout the paper. For more details about this topic, we refer the reader to the manuscripts [24,25].
A graph is any pair G = (V(G), E(G)) formed by a set V(G) of vertices and a set E(G) of edges so that each edge joins two vertices, which are then said to be adjacent. From now on, let vw be the edge formed by two vertices v, w ∈ V(G). If v = w, then the edge constitutes a loop. A graph is called simple if it does not contain loops. Further, the number of vertices of a graph is its order. A graph is called finite if its order is finite. This paper deals with the direct product G × H of two finite and simple graphs G = (V(G), E(G)) and H = (V(H), E(H)). Its vertex set is the Cartesian product V(G) × V(H). Two vertices (v, v ) and (w, w ) in such a set are adjacent if and only if vw ∈ E(G) and v w ∈ E(H).  The set of vertices that are adjacent to a vertex v ∈ V(G) constitutes its neighborhood N G (v). The cardinality d G (v) of this set is the degree of the vertex v. If there is no risk of confusion, then we use the respective notations N(v) and d(v). Furthermore, we denote, respectively, δ(G) and ∆(G) the minimum and maximum vertex degree of the graph G. The following result follows straightforwardly from the previous definitions. Lemma 1. Let G and H be two finite simple graphs. Then, A finite graph is called complete if all its vertices are pairwise adjacent. A path between two distinct vertices v and w of a given graph G is any ordered sequence of adjacent and pairwise distinct vertices v 0 = v, v 1 , . . . , v n−2 , v n−1 = w in V(G), with n > 2. If v = w, then such a sequence is called a cycle. A graph is said to be connected if there always exists a path between any pair of vertices. Further, if all the vertices of a cycle are joined to a new vertex, then the resulting graph is called a wheel. Such a new vertex is called the center of the wheel graph. From here on, let K n , P n , C n and W n respectively denote the complete graph, the path, the cycle and the wheel graph of order n.
A proper k-coloring of a graph G is any map c : V(G) → {0, . . . , k − 1} assigning k colors to the set of vertices V(G) so that no two adjacent vertices have identical color. The minimum positive integer k for which such a proper k-coloring exists is the chromatic number χ(G) of the graph G. Particular cases of proper coloring and chromatic number are the so-called r-dynamic proper k-coloring and the r-dynamic chromatic number, which have already been described in the introductory section (see (1.1)). The following results are known.

Lemma 2.
[1] Let G be a graph and let r be a positive integer. Then,

Lemma 3.
[3] Let n and r be two positive integers. Then, the following results hold. a) If n > 2, then χ r (P n ) = In case of dealing with the r-dynamic chromatic number of a direct product of graphs, the following results hold.

Lemma 5.
[23] Let G and H be two finite simple graphs and let r be a positive integer such that r ≤ δ(G ), for some G ∈ {G, H}. Then,

Theorem 6.
[23] Let m ,n and r be three positive integers such that m, n > 2. Then,

Dynamic coloring of the direct product of a path and a complete graph
In this section, we study the r-dynamic chromatic number of the direct product of a path P m = u 0 , . . . , u m−1 and a complete graph K n of set of vertices where m and n are two positive integers such that m > 2. From Lemma 1, we have that δ(P m × K n ) = n − 1 ≤ 2(n − 1) = ∆(P m × K n ).
More specifically, for each vertex (u i , v j ) ∈ P m × K n , it is Firstly, we focus on the case n ≤ 3, which follows readily from already known results.
Theorem 7. Let m and r be two positive integers such that m > 2. Then, the following assertions hold.
Proof. The first assertion follows simply from the fact that the direct product P m × K 1 is not connected. Further, the second assertion follows from Lemma 3 once it is observed that the direct product P m × K 2 is formed by a pair of disjoint paths of order m. Finally, the third assertion follows from Theorem 6 once it is observed that the complete graph K 3 constitutes a cycle of order three. Now, let us prove a pair of preliminary lemmas that are useful to deal with the case n > 3. In order to simplify the notation, for each given proper coloring c of the direct product P m × K n , we denote from now on c i, j := c(u i , v j ), for all non-negative integers i < m and j < n.
Lemma 9. Let m, n and r be three positive integers such that m > 2, n > 3 and 3n Proof. Since r ≤ 2n − 2 = ∆(P m × K n ), we have from Lemma 2 that r + 1 ≤ χ r (P m × K n ). Let us suppose the existence of an r-dynamic proper (r + 1)-coloring c of the direct product P m × K n .
Similarly to the proof of Lemma 8, we can suppose, without loss of generality, that c 1, j = j, for all non-negative integer j < n, and c 0, j = n + j, for all non-negative integer j < r − n. Then, since |c(N((u 1 , v n−1 ))) \ {n, . . . , r − 1}| = n and the map c is a proper coloring, it must be c 2, j = j, for all non-negative integer j < r − n, and j ∈ {c 0, j , c 2, j }, for all j ∈ {r − n, . . . , n − 1}. But then, since n + j must be a color in the set c(N((u 1 , v j ))), for all non-negative integer j < r − n, it should also be n + j ∈ {c 0,r−n , . . . , c 0,n−1 , c 2,r−n , . . . , c 2,n−1 }, for all non-negative integer j < r − n. It is only possible if and only if r − n + 1 ≤ 2n − r − 1. That is, it should be r ≤ 3n 2 − 1, which contradicts the hypothesis. The following result establishes the r-dynamic chromatic number of the direct product of a path and a complete graph of order n > 3.
Theorem 10. Let m, n and r be three positive integers such that m > 2 and n > 3. Then, otherwise.
Proof. Let us study separately each case by defining an appropriate r-dynamic proper coloring c of the corresponding direct product P m × K n satisfying Condition (1.1).
Then, let the map c be defined so that Condition (1.1) holds and hence, χ r (P m × K n ) = r + 2. Figure 2 illustrates the direct product P 3 × K 5 , for r = 3. • Case n − 1 ≤ r < 3n 2 . From Lemma 2, we have that r + 1 ≤ χ r (P m × K n ). Then, let the map c be defined recursively as follows.
-For each non-negative integer j < n, we have that c 0, j = j.
-For each positive integer i < m and each non-negative integer j < n, we have that Condition (1.1) holds and hence, χ r (P m × K n ) = r + 1. Figures 3-5 illustrate the direct product P 6 × K 5 , for r ∈ {4, 5, 6}.   • Case 3n 2 ≤ r ≤ 2n − 2. From Lemma 9, it is r + 2 ≤ χ r (P m × K n ). Then, let the map c be defined so that, for each pair of non-negative integers i < m and j < n, we have that Condition (1.1) holds and hence, χ r (P m × K n ) = r + 2. Figure 6 illustrates the direct product P 5 × K 7 , for r = 10. • Case r > 2n − 2.
The result follows from the previous case and Lemma 2, once we notice that ∆(P m × K n ) = 2n − 2.

Dynamic coloring of the direct product of a path and a wheel graph
Let m and n be two positive integers such that m > 2 and n > 3. In this section, we study the r-dynamic chromatic number of the direct product of a path P m = u 0 , . . . , u m−1 and a wheel graph W n of set of vertices Here, v denotes the center of the wheel graph. Thus, W n contains the cycle graph From Lemma 1, we have that Firstly, let us prove a result that enables us to focus on those direct products P m × W n such that either n = 7, or n is even, or n = 4t + 1, for some t ≥ 1.
Lemma 11. Let m > 2, n > 3 and r be three positive integers such that n is odd. Then, Proof. The result follows straightforwardly from the fact that the direct product P m × W 2n+1 may be considered as two direct products P m ×W n+1 , whose sets of vertices are disjoint except for those vertices corresponding to the common center of their respective wheel graphs.
The following preliminary lemmas establish certain bounds for the r-dynamic chromatic number χ r (P m × W n ). In order to simplify the notation, for each given proper coloring c of the direct product P m × W n , we denote from now on c i, j := c(u i , v j ), for all non-negative integers i < m and j < n − 1. In addition, all the indices of the vertices v j associated to the wheel graph W n are considered to be taken modulo n−1. Let us start with a pair of bounds of the r-dynamic chromatic number χ r (P m ×W n ) arising from the fact that every wheel graph contains a cycle.
Lemma 12. Let m, n and r be three positive integers such that m > 2, n > 3 and r > 1. Then, whenever there exists an (r − 1)-dynamic proper χ r−1 (P m × C n−1 )-coloring c of the direct product P m × C n−1 such that the following two conditions hold.
Proof. Let us suppose the existence of the map c in the hypothesis. Then, let c : V(P m × W n ) → {0, . . . , χ r−1 (P m × C n−1 )} be defined so that, for each non-negative integer i < m, we have that c((u i , v)) = χ r−1 (P m × C n−1 ), and c i, j = c i, j , for all non-negative integer j < n − 1. It is simply verified that this map c is a proper coloring of the direct product P m × W n satisfying Condition (1.1), for every vertex (u i , v j ) ∈ V(P m × W n ). Moreover, the compliance of both conditions (a) and (b) enable us to ensure that Condition (1.1) also holds for every vertex (u i , v) ∈ V(P m × W n ), and hence, the result holds. Figure 7 illustrates this constructive proof for the direct products P 4 × C 5 and P 4 × W 6 . Let us establish now a series of lower bounds of χ r (P m × W n ), for r ≥ 2.
In a recursive way, it is simply verified that {c i, j : 0 ≤ j < 4} = 4 and c((u i , v)) = c((u 0 , v)), for all non-negative integer i < m. Thus, the map c : V(P m × C 4 ) → {0, 1, 2, 3, 4} \ {c((u 0 , v))} that is defined so that c i, j = c i, j , for all non-negative integers i < m and j < 4 is a 4-dynamic proper 4-coloring of the direct product P m × C 4 . It contradicts Theorem 6 and hence, the result holds. • Case n = 6.
It follows simply from the case n = 6 and Lemma 11. Then, 6 < χ 5 (P m × W n ).
Further, in order to establish an upper bound based on Lemma 5, the following proposition determines the 3-dynamic chromatic number of a wheel graph.
Proposition 18. Let m and n be two positive integers such that m > 2 and n > 3. Then, Proof. Since δ(W n ) = 3, for all n > 3, Lemma 5 implies that χ 3 (P m × W n ) ≤ χ 3 (W n ). In addition, from Lemma 2, we have that 4 ≤ χ 3 (W n ). Let us study separately each case by describing to this end an appropriate 3-dynamic proper coloring c of the wheel graph W n satisfying Condition (1.1). An illustrative example of each case is shown in Figure 8.
Let the map c be defined so that c(v) = 3 and c(v j ) = j mod 3, for all non-negative integer j < n−1.
The result holds because, from Condition (1.1), no two vertices in W n can share the same color in any given 3-dynamic proper coloring of the wheel graph W 6 . An illustrative example is shown in Figure 8. After enumerating all the previous bounds, we are in condition of determining exactly the r-dynamic chromatic number of the direct product P m × W n . Firstly, let us establish the r-dynamic chromatic number of the direct product P m × W 4 , which follows readily from Theorem 10, once we notice that the wheel graph W 4 coincides with the complete graph K 4 .
Theorem 19. Let m and r be two positive integers such that m > 2. Then, otherwise.
The following theorem is the main result of the section. It establishes the r-dynamic chromatic number of the direct product of a path and a wheel graph of order n 4.
Theorem 20. Let m, n and r be three positive integers such that m > 2 and n > 4. Then, if r = 2 and n is odd, Proof. Let us study separately each case by defining to this end an appropriate r-dynamic proper coloring c of the direct product P m × W n satisfying Condition (1.1).
• Case r = 2. From Lemma 2, we have that 3 ≤ χ 2 (P m × W n ). Moreover, if n is even, then Lemma 13 implies that 4 ≤ χ 2 (P m × W n ). The result holds because, since r = 2 < 3 = δ(W n ), we have from Lemmas 4 and 5 that 3, if n is odd, 4, if n is even.
Finally, since the 2-dynamic proper 4-coloring of the direct product P m × C 5 that is described in the proof of Theorem 17 in [23] satisfies both conditions (a) and (b) of Lemma 12, this last result enables us to ensure that χ 3 (P m × W 6 ) ≤ 5 and hence, that this upper bound is reached.
From Lemma 2, we have that 5 ≤ χ 4 (P m × W n ). Then, the following study of cases arise.
-Subcase n = 6k + 1, for some k > 0. Let the map c be defined so that c((u i , v)) = 4, for all non-negative integer i < m, and such that the following assertions hold. * c i, j = c i, j mod 6 , for all non-negative integers i < m and j < n. * Let j < 6, l < 6 and t < m 6 be three non-negative integers such that 6t + j + l < m. Then, Condition (1.1) holds and hence, χ 4 (P m × W n ) = 5. Figure 9 illustrates the direct product P 7 × W 7 . -Subcase n = 6k + 4, for some k > 0. Let the map c be defined so that, for each non-negative integer i < m, we have that c((u i , v)) = 4, and Condition (1.1) holds, and hence, χ 4 (P m × W n ) = 5. Figure 10 illustrates the direct product P 17 × W 16 . -Subcase n ∈ {5, 6, 11}.
From Lemma 15, we have that 6 ≤ χ 4 (P m × W n ). In order to prove the case n = 5, let the map c be defined so that the following assertions hold. * c i, j = j, for all non-negative integers i < m and j < n − 1. * For each non-negative integer i < m, we have that Condition (1.1) holds, and hence, χ 4 (P m × W 5 ) = 6. Figure 11 illustrates the direct product P 4 × W 5 . Further, both cases n = 6 and n = 11 follow from Lemma 12 and the respective 4-dynamic proper 5-colorings that are described in the proof of Theorem 17 in [23].
• Case r = 5. From Lemma 2, we have that 6 ≤ χ 5 (P m × W n ). Then, Lemmas 11 and 16 enable us to focus on the following study of cases.
From Lemma 16, we have that 7 ≤ χ 5 (P m × W 5 ). Then, let the map c be defined so that otherwise.
-Subcase n = 4t + 1, for some t ≥ 2. From Lemma 16, we have that 7 ≤ χ 5 (P m × W n ). Let the map c be defined so that In addition, for each (u i , v j ) ∈ V(P m × W n ), we have that  -Subcase n = 7. Again from Lemma 16, we have that 7 ≤ χ 5 (P m × W 5 ). Then, let the map c be defined so that c(u i , v) = 6, for all positive integer i < m, and Condition (1.1) holds, and hence, χ 5 (P m × W 7 ) = 7. Figure 14 illustrates the direct product P 6 × W 7 . Figure 14. 5-dynamic proper 7-coloring of the direct product P 6 × W 7 .
Let the map c be defined so that, for each non-negative integer j < n − 1, we have that otherwise.
Let c be the map defined in the previous subcase (n = 6t + 2). Then, let the map c be defined so that, for each pair of non-negative integers i < m and j < n − 1, we have that Condition (1.1) holds, and hence, χ 6 (P m × W n ) = 8. Figure 22 illustrates the direct product P 6 × W 16 . • Case 7 ≤ r ≤ 2(n − 1). From Lemma 17, we have that r + 2 ≤ χ r (P m × W n ). The following study of cases arises. In all of them, the described map c satisfies that r + 1, otherwise.
Let t < r be such that n − 1 ≡ t (mod r). Then, let the map c be defined so that k, if i = 0 and j = 2k + l, for some k < t and l ∈ {0, 1}, j − t, if i = 0 and 2t ≤ j, (c i−1, j − 2) mod r, otherwise.
Condition (1.1) holds, and hence, χ r (P m × W n ) = r + 2. Figure 23 illustrates the direct product P 5 × W 12 , for r = 8. -Subcase n + 2 ≤ r. Let c be the map just described in the previous subcase (r ≤ n − 1). Then, let the map c be defined so that 2} and c i, j < r − n + 1, c i, j , otherwise.

Conclusion and further works
This paper has delved into the study of the r-dynamic chromatic number of the direct product of two given graphs. More specifically, it has explicitly been determined the r-dynamic chromatic number of the direct product of any given path P m with either a complete graph K n or a wheel W n . In this regard, Theorems 10, 19 and 20 are the main results of the manuscript. Particularly, it has been obtained that r + 1 ≤ χ r (P m × G) ≤ 2n, for all G ∈ {K n , W n }. In addition, we have also established in Proposition 18 the 3-dynamic chromatic number of any wheel graph.
Similarly to the previous work of the authors in the topic [23], a significant number of technical results is required in order to prove the main theorems of the paper. This fact enables us to corroborate that the problem of r-dynamic coloring the direct product of two given graphs is not trivial at all. Of particular interest for the continuation of this paper is the study of the r-dynamic coloring of the direct product of two complete graphs and that one concerning the direct product of two wheel graphs. The r-dynamic coloring of the direct product of a path, a complete graph or a wheel with other types of graphs is also established as related further work.