GRACEFUL CHROMATIC NUMBER OF SOME CARTESIAN PRODUCT GRAPHS

: A graph G ( V,E ) is a system consisting of a ﬁnite non empty set of vertices V ( G ) and a set of edges E ( G ). A ( proper ) vertex colouring of G is a function f : V ( G ) → { 1 , 2 ,...,k } , for some positive integer k such that f ( u ) 6 = f ( v ) for every edge uv ∈ E ( G ). Moreover, if | f ( u ) − f ( v ) | 6 = | f ( v ) − f ( w ) | for every adjacent edges uv,vw ∈ E ( G ), then the function f is called graceful colouring for G . The minimum number k such that f is a graceful colouring for G is called the graceful chromatic number of G . The purpose of this research is to determine graceful chromatic number of Cartesian product graphs C m × P n for integers m ≥ 3 and n ≥ 2, and C m × C n for integers m,n ≥ 3. Here, C m and P m are cycle and path with m vertices, respectively. We found some exact values and bounds for graceful chromatic number of these mentioned Cartesian product graphs.


Introduction
A graph G(V, E) is a system consisting of a finite non empty set of vertices V (G) and a set of edges E(G).Let G and H be two disjoint graphs.The Cartesian product of G and H, denoted by G × H, is the graph with vertex set V (G) × V (H), and edges xy, uv ∈ V (G) × V (H) are adjacent in G × H, if x = u and yv ∈ E(H) or y = v and xu ∈ E(G).A (proper) vertex colouring of G is a way of colouring vertices in G such that each adjacent vertices are assigned to different colours.
If for a vertex colouring of G we have that every adjacent edges in G have different induced colours, then the vertex colouring is called graceful.We may think a graceful colouring of G as a function f : V (G) → {1, 2, . . ., k}, for some positive integer k, such that for every edge uv ∈ E(G) we have f (u) = f (v), and for any vertex u ∈ V (G) we have |f (u) − f (v)| = |f (u) − f (w)| for every vertices v, w ∈ V (G) which are adjacent to u.The absolute value |f (u) − f (v)| for every uv ∈ E(G), is the induced label of the edge uv ∈E(G).In this sense, the terms colour and label are interchangeable.The smallest value of k for which the function f is a graceful vertex colouring of G is called the graceful chromatic number of G.The graceful colouring is a variation of graceful labeling which was introduced by Alexander Rosa in 1967 (see Gallian in [5]).Whereas, the notion of graceful colouring was introduced by Gary Chartrand in 2015, as a variant of the proper vertex k-colouring problem (see [3]).Since then, researches on graceful colouring numbers started to be celebrated.
Byers in [3] derived exact values for the graceful chromatic number of some graphs: path, cycle, wheel, and caterpillar; and introduced some bounds for certain connected regular graphs.Moreover, English, et al. in [4] invented graceful chromatic number of some classes of trees, and gave a lower bound for the graceful chromatic number of connected graphs with certain minimum degree.Mincu et al. in [6] derived graceful chromatic number of some well-known graph classes, such as diamond graph, Petersen graph, Moser spindle graph, Goldner-Harary graph, friendship graphs, and fan graphs.Graceful chromatic number of some particular unicyclic class graphs were presented by Alfarisi et al. (2019) in [1].
Furthermore, in 2022, Asy'ari et al. in [2] presented graceful chromatic numbers of several types of graphs, including star graphs, diamond graphs, book graphs.In addition, Asy'ari, et al. also stated some open problems.One of the problems is to determine the graceful chromatic number of some Cartesian product of certain graphs.Here we derive graceful chromatic number of Cartesian product graph C m × P n , m ≥ 3, n ≥ 2, where C m is the cycle with m vertices and P n is the path with n vertices.The Cartesian product graph C m × P n is known as prism for n = 2 and as generalized prism for n ≥ 3. We also introduce bounds for Cartesian product graph To proceed with the main results, we need to introduce some introductory facts which will be beneficial for our further discussion.
Let G be a graph and x be a vertex of G.All vertex which are adjacent to x are called the neighbors of x, and denoted by N (x).The degree of the vertex x, denoted by deg(x), is equal to the cardinality of N (x), deg(x) = |N (x)|.We will start with the following lemma.
Lemma 1.Let G be a graph and u be a vertex in G with degree the smaller possible colours we can assign for the all d neighbors v ∈ N (u) of u, are 2, 3, . . ., d and the colour d + 1.This means that, there is a vertex v ∈ N (u) with f (v) ≥ d + 1 = d + a.We are done for the case a = 1.Now, assume f (u) = a, 1 < a ≤ d.Note that the colours k and 2a−k, for every k, 1 ≤ k ≤ a−1, can not be assigned simultaneously for the vertices in N (u), since they give the same difference from the colour a.Therefore, the maximum number of colours we may assign from the first 2(a − 1) smallest colours {k, 2a − k : 1 ≤ k ≤ a − 1} is equal to a − 1.It implies that the remaining vertices in N (u) which are not coloured yet, is at least d − (a − 1) vertices.The colours we need for these vertices are started from a colour ≥ 2a.This means that the next d − (a − 1) smallest colours we should assign are 2a, 2a + 1, . . ., 2a + (d − (a − 1) − 1).So, there is a vertex v ∈ N (u) such that its colour In a specific case, the colour of a vertex u is equal to the degree of u, f (u) = deg(u), we have the following corollary.The following result was introduced by Byers (2018) in [3].
Lemma 2 (Byers in [3]).The graceful chromatic number of cycle C n on n ≥ 3 vertices is Then, we will introduce some terminologies related with certain ladder graphs.
A ladder of 2m vertices, m ≥ 2, denoted by L m , is the Cartesian product graph of the path on m vertices and the path on two vertices.The ladder L 2 is the cycle graph of four vertices.Assume that the vertices of L m are v 1 , v 2 , . . ., v m , w 1 , w 2 , . . ., w m such that its edges are On the other side, let C m × P 2 , m ≥ 3, be a prism.This prism has vertex set {v 1 , v 2 , . . ., v m , w 1 , w 2 , . . ., w m } and edge set After opening C m × P 2 about the edge v 1 w 1 into the ladder L m+1 , the vertices v 1 and w 1 copy themselves into two copies each; the first copy of v 1 (resp.w 1 ) is adjacent with v 2 (resp.w 2 ), and the second copy of v 1 (resp.w 1 ) is adjacent with v m (resp.w m ).These last vertex copies in the ladder L m+1 are named as v m+1 and w m+1 , respectively.Therefore, if f a colouring for the prism C m × P 2 , then in the ladder L m+1 we have f (v 1 ) = f (v m+1 as well as f (w 1 ) = f (w m+1 ).In this case, we may also call C m × P 2 as the closed graph of L m+1 about the edges v 1 w 1 and v m w m .
In the following lemma we will show that a ladder of 2m vertices, with m ≡ 0 (mod 4), can not be gracefully coloured using 4 colours.
Observe that the colour of v j (resp.w j ) must be the same with the colour of w j+2 (resp.v j+2 ) or of w j−2 (resp.v j−2 ) for realizable integer j (realizable means in the range of discussion).Without loss of generality, let the colour of v 1 is a.Therefore, the colour of w 4s+3 and of v 4t+1 is a, for some realizable non-negative integers s, t.Now let us see cases: r = 1, r = 2, and r = 3. Suppose that f is a graceful colouring for This implies a contradiction.So, for r = 1 the graph C m × P 2 can not be gracefully coloured.
Case r = 2. Applying a similar argument, by assuming the colour of v 1 is a, we have that In graph C m × P 2 , vertices w 1 and w m+1 are identical.On the other side, w 1 is adjacent with v 1 , so that they can not get the same colour.Thus, a contradiction occurs.
Case r = 3. Again by using a similar reason, we have that f (w m ) = f (w 4k+3 ) = f (v 1 ) = a.We know that w m+1 in C m × P 2 is identified with w 1 , and therefore is adjacent with both w m and v 1 .This implies that the induced edge colours of v 1 w 1 (= v 1 w m+1 ) and w 1 w m are the same which then contradicts the gracefulness property.
In any case we have proven that C m × P 2 , m ≡ 0(mod 4), can not be gracefully coloured using only 4 colours.

Results on prism and generalized prism graphs
In this section, we will be dealing with the graceful chromatic number of prism C m × P 2 first, m ≥ 3, and then with the graceful chromatic number of generalized prism graphs C m × P n , m, n ≥ 3.As for some consequences, we also derive some bounds for graceful chromatic number of graph C m × C n , m, n ≥ 3, for some specific values of m and n.
Our main discussion will be separated into two subsections: For C m ×P 2 , m ≥ 3 and for C m ×P n , with m, n ≥ 3.

Prism graph
Theorem 1.If m ≡ 0 (mod 4), then the graceful chromatic number of graph C m × P 2 is equal to 5. P r o o f.Note that the graph C m × P 2 contains subgraph C 4 .Based on Lemma 2, we may conclude that χ g (C m × P 2 ) ≥ 4. Since all vertices of C m × P 2 has degree 3, if the colour 3 is used, then by Corollay 1, the colour greater than 6 should occur.Therefore, the four colours we will use are 1, 2, 4, and 5. Now we will prove that using these four colours, we are able to colour C m × P 2 gracefully.To confirm this, we will do by introducing the following graceful colouring technique for C m × P 2 using only labels 1, 2, 4, and 5.
Let the vertices of C m × P 2 is the set and its edge set is Define a colouring f for C m × P 2 as follows. (2.1) Based on the above function f , it is clear that for every adjacent vertices u and v we have f (u) = f (v).We can immediately observe that for any adjacent edges uw and wv in C m we have Furthermore, we also have Remember that each vertex u in C m × P 2 has degree 3; say x 1 , x 2 , and x 3 are the vertices adjacent to u.From the function f we can immediately conclude that the set is equal to {1, 2, 3} or to {1, 3, 4}.Thus, the function f satisfies the property to become graceful colouring for C m × P 2 .Therefore, χ g (C m × P 2 ) = 5.
Theorem 2. If m ≡ 0 (mod 4), then the graceful chromatic number of graph C m × P 2 is equal to 6. P r o o f.The proof of Theorem 2 will make use of the result described in the proof of Theorem 1.
For some positive integer k ≥ 1, consider C 4k × P 2 which is coloured as in (2.1).Let the ladder L 4k+1 be the open graph of C 4k × P 2 about v 1 w 1 .Since C m × P 2 contains subgraph C 4 , to colour it gracefully, one needs at least 4 colours.But, when m ≡ 1, 2 or 3 (mod 4), based on Lemma 3, we can not colour the graph C 4k × P 2 gracefully using only 4 colours.Therefore, we have to use at least 5 colours.The smallest five colours are 1, 2, 3, 4, and 5. But, based on Corollary 1, whenever we apply 3 for a vertex colour, the colour 6 or greater colour must occur.Thus, the graceful chromatic number of C m × P 2 is at least 6.To conclude that χ g (C m × P 2 ) = 6, we will proceed by showing that a graceful colouring exist with maximum colour 6, as follows.
. Now, we colour vertices using the following function f : The coloured C 5 ×P 2 will be used as the seed of our general construction for Case 1, and its diagram is depicted in Fig. 2.
Consider the opened ladder L 6 from the coloured C 5 × P 2 above about a 1 b 1 .In L 6 , the colours of a 1 , a 2 , a 3 , a 4 , a 5 , and a 6 are 1, 2, 5, 3, 4, and 1, while the colours of b Then, consider the open ladder L 4k+1 , for some positive integer k ≥ 1, from the coloured C 4k × P 2 in Theorem 1 about v 1 w 1 .Here, the colours of v 1 and w 1 are also 1 and 5, respectively.The same colours are also for v 4k+1 which is 1, and for w 4k+1 which is 5. Based on (2.1), we have f (v 4k ) = 2, and f (w 4k ) = 4.By identifying v 4k+1 with a 6 and w 4k+1 with b 6 , and maintaining the other vertex colours, then we get a new ladder on 4(k + 1) + 2 vertices, L 4(k+1)+2 , with graceful colouring.
From here, we may infer that the graceful chromatic number of the graph C m × P 2 , for m ≡ 1 (mod 4) is equal to 6.
As a seed graph, we define the following colouring for C 6 × P 2 as follows.
By inspection we can verify that the above colouring for C 6 × P 2 is graceful.The diagram of the coloured graph is shown in Fig. 3.
Let the ladder of 7 vertices, L 7 , is the open graph from the C 6 × P 2 above about v 1 w 1 .We emphasize here that in this ladder L 7 , vertices a 7 and b 7 have colours 1 and 5, respectively; the same as the colours of a 1 and b 1 , respectively.
We use again the same ladder L 4k+1 , k ≥ 1, as in Case 1.Now we identify v 4k+1 with a 7 and w 4k+1 with b 7 , and maintaining the other vertex colours.Then we get a new ladder on 4(k + 1) + 3 vertices, L 4(k+1)+3 , with graceful colouring.
Furthermore, we identify v 1 with a 1 and w 1 with b 1 in the ladder L 4(k+1)+3 .Based on the previous colours, we know that the colours of v 2 , w 2 , a 2 , b 2 , v 1 = a 1 , w 1 = b 1 , are 4, 2, 3, 6, 1, 5, respectively.This means that after the last identification, the gracefulness colouring of C 4(k+1)+2 are maintained.Thus, we may conclude that C 4(k+1)+2 × P 2 is with graceful colouring.A graceful labeled C 10 × P 2 which is constructed using this method is depicted in Fig. 4. From here, we may infer that the graceful chromatic number of the graph C m × P 2 , for m ≡ 2 (mod 4) is equal to 6.
Case 3 : m ≡ 3 (mod 4).Here we will introduce a construction for graceful colouring of C m × P 2 with m ≡ 3 (mod 4).We start with We can immediately check that this colouring f is graceful.The diagram of the gracefully coloured graph C 3 × P 2 is shown in Fig. 5.We can verify that the graceful chromatic number of this graph is 6.
We should mention again that this above colouring of C 3 × P 2 is graceful.As we did for Case 1 and Case 2, first we will observe the open ladder L 4 from C 3 × P 2 about a 1 b 1 .In this L 4 , the  Now we identify v 4k+1 with a 4 and w 4k+1 with b 4 to obtain a graceful colouring ladder L 4k+4 .Let us denote the colouring as α.We can easily see that in this ladder we have α(a 1 ) = α(v 1 ) = 1 and α(b 1 ) = α(w 1 ) = 5.Moreover, we have also α(a and α(w 2 ) = 2. Thus, by identifying v 1 with a 1 and w 1 with b 1 , we get a graceful colouring C 4k+3 × P 2 , with graceful chromatic number is 6.See the labeled graph C 7 × P 2 in Fig. 6 as an example of the graph resulted from the construction.
Therefore, we may conclude that the graceful chromatic number of the graph C m × P 2 , with m ≡ 3 (mod 4) is also 6.
Since in all cases of m we proved that C m × P 2 has graceful chromatic number 6, we may conclude that χ g (C m × P 2 ) = 6.

Results on generalized prism graphs
For a graph G, let f be a graceful colouring for G.It is obvious that for a vertex u ∈ V (G), if v, w ∈ N (u), then f (v) = f (w).Therefore, we can immediately observe that the graph P 3 × P 3 can not be coloured by only four different colours.This observation gives χ g (P 3 × P 3 ) ≥ 5.
But, if we use only five colours 1, 2, 3, 4 and 5, the center vertex of P 3 × P 3 must be 1 or 5.Then, by inspection we can show that using only five colours, we can not colour P 3 × P 3 gracefully.This gives the following lemma.Lemma 4. The graceful chromatic number of the graph P 3 × P 3 , χ g (P 3 × P 3 ) ≥ 6.
The following Lemma 5 will be an important tool for the proofs of our main results encountered in this section.
Let the six colours be 1, 2, 3, 4, 5 and 6.Based on Lemma 1, since the degree of vertices v 11 and v 21 each is four, the colours 3 and 4 both can not be used for these two vertices.So, there are four colours: 1, 2, 5, and 6 that can be assigned for the vertices v 11 and v 21 .In total, there are six different combinations for colouring these two vertices: {α(v 11 ), α(v 21 )} = {a, b}, a, b ∈ {1, 2, 5, 6}, with a = b.We can check by inspection that any one of these combinations results in the colouring α is not graceful.But, for the space consideration, we will only describe the detail process for combination {α(v 11 ), α(v 21 )} = {1, 2} as in Fig. 8.Note that the case α(v 11 ) = a and α(v 21 ) = b is similar to the case α(v 11 ) = b and α(v 21 ) = a.
The explanation of the colouring process in Fig. 8 is the following: 1) The colours α(v 11 ) = 1 and α(v 21 ) = 2 are fixed as the initial combination.
2) The next vertex colouring follows the following vertices order: v 20 , v 10 , v 00 , v 01 , v 02 , v 12 , v 22 .Note that α(v 3j ) := α(v 0j ), ∀j = 0, 1, 2, based on the restriction imposed for α. 3) For some colours x, y and z, a notation x/y/z means that we assign the colour y(indicated with bold face) for the related vertex among the possible colours x, y and z. 4) The colour which stands alone (written in red bold face), indicates that the colour is the only possible colour for the related vertex.5) The red cross sign X informs that the colouring process is discontinue at the related vertex, since there is no possible choice of colours to colour the vertex.The appearance of X indicates that the colouring fails to be graceful.
From Fig. 8 we can see that each colouring process ends to be not graceful which is indicated by the appearance of the sign X.Thus, we may conclude that under the restriction α(v 1j ) = α(v 4j ), j = 0, 1, 2, using exactly six different colours, we can not colour the graph P 4 × P 3 gracefully.If we extend this last observation to graph P 4 × P n , n ≥ 3, with under restriction that α(v 0j ) = α(v 3j ), j = 0, 1, . . ., n − 1, we may also conclude that we need at least seven colours to maintain the colouring α becomes graceful for P 4 × P n .From this last observation we can formulate the following result.Lemma 6.For n ≥ 3, the graceful chromatic number of the graph C 3 × P n , χ g (C 3 × P n ) ≥ 7.
An example of a graceful coloured graph C 6 × P n using (2.2) is shown in Fig. 9.In this figure we may also see the related open graph L 7,n of C 6 × P n .Corollary 2. For any positive integers m, n ≥ 3, with m ≡ 0 (mod 3) and with n ≡ 0 (mod 6), χ g (C m × C n ) = 7. P r o o f.The proof of this corollary may be derived from (2.2).From Theorem 3 we conclude that χ g (C m × P n ) = 7, if m ≡ 0 (mod 3), and n ≥ 3. From (2.2) we know that f (v ij ) = f (v ik ) whenever |j − k| ≡ 0 (mod 6).Thus, if n ≡ 0 (mod 6), then if we identify vertex v i0 and v in for every i, 0 ≤ i ≤ m − 1 in C m × P n , then we get a graceful coloured graph C m × C n , m ≡ 0 (mod 3) and n ≡ 0 (mod 6).Therefore, we may conclude that χ g (C m × C n ) = 7 where m ≡ 0 (mod 3) and n ≡ 0 (mod 6).
In the remaining part of this section we will see the graceful colouring number for C m × P n , with m ≡ 0 (mod 3), n ≥ 3. We start to observe the case m ≡ 1 (mod 3) as we formulate in the following theorem.Theorem 4. If m ≡ 1 (mod 3), then 7 ≤ χ g (C m × P n ) ≤ 8. P r o o f.We will make use of prism graph C 4 × P n as the seed of our graceful colouring construction.We first introduce a colouring for the graph C 4 × P n , n ≥ 3. 7 ≤ χ g (C 5 × P n ) ≤ 8. Similar to the previous corollaries, here we formulate the following corollary as a consequence of Theorem 5.

Conclusion
In the discussion above, it has been proven that prism graph C m × P 2 has a chromatic number equal to 5 when m ≡ 0 (mod 4), and equal to 6 when m ≡ 0 (mod 4).While for generalized prism C m × P n we found that its chromatic number is equal to 7 while m ≡ 0 (mod 3).Whereas for m ≡ 0 (mod 3), we got that 7 ≤ χ g (C m × P n ) ≤ 8. Based on these results, we could also derive some exact and bound values of graceful chromatics number of C m × C n for certain m, n ≥ 3. Regarding this last observation, we propose the following open problem and conjecture.

Corollary 1 .
In a graph G with graceful colouring f , if the vertex u has degree d ≥ 1 and colour d, then there is a vertex v ∈ N (u) with colour f (v) ≥ 2d.P r o o f.Let G be a graph and u be a vertex of G with deg(u) = d.Let f be a graceful colouring for G where f (u) = d.By Lemma 1, we found a neighbor v of u such that f (v) ≥ d + d = 2d.
For m ≥ 4, if the vertices v 1 and v m , and the vertices w 1 and w m are identified, then we obtain a prism C m−1 × P 2 .In this resulting C m−1 × P 2 , v 1 = v m , w 1 = w m , and edge v 1 w 1 = v m w m .Due to this, we may call the ladder L m as the open graph of C m−1 × P 2 about the edge v 1 w 1 .

Lemma 3 .
Using four different colours, the graph C m × P 2 , with m ≥ 3, m ≡ 0 (mod 4), can not be gracefully coloured.P r o o f.Let a, b, c and d be four different colours, and let m = 4k + r, 1 ≤ r ≤ 3. Consider the ladder L m+1 as the opened graph of C m × P 2 .Let the vertex and edge sets of the ladder L m+1 be {v i , w i : 1

Fig. 9
Fig.9also helps us to be able to check by inspection that f is a graceful colouring for the graph C m × P n , with m ≡ 0 (mod 3).Therefore, we may conclude that this graph has chromatic number 7.