CC BY
3URAL MATHEMATICAL JOURNAL, Vol. 9, No. 2, 2023, pp. 193-208
DOI: 10.15826/umj.2023.2.016
GRACEFUL CHROMATIC NUMBER OF SOME CARTESIAN
PRODUCT GRAPHS1
I Nengah Suparta"", Mathiyazhagan Venkathacalam6"", I Gede Aris Gunadi"""",
Putu Andi Cipta Pratama"""""
"Department of Mathematics, Universitas Pendidikan Ganesha, Jl. Udayana 11, Singaraja-Bali 81117, Indonesia
^Department of Mathematics, Kongunadu Arts and Science College, Coimbatore-641029, Tamil Nadu, India
"nengah.suparta@undiksha.ac.id ""venkatmaths@kongunaducollege.ac.in """igedearisgunadi@undiksha.ac.id """"andicipta25@gmail.com
Abstract: 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) = f(v) for every edge uv € E(G). Moreover, if |f(u) — f (v)| = |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 Cm X Pn for integers m > 3 and n > 2, and Cm X Cn for integers m, n > 3. Here, Cm and Pm 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.
Keywords: Graceful colouring, Graceful chromatic number, Cartesian product.
1. 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 x H, is the graph with vertex set V(G) x V(H), and edges xy,uv € V(G) x V(H) are adjacent in G x 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.
1This work was supported by LP2M of Universitas Pendidikan Ganesha.
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 Cm x Pn, m > 3,n > 2, where Cm is the cycle with m vertices and Pn is the path with n vertices. The Cartesian product graph Cm x Pn is known as prism for n = 2 and as generalized prism for n > 3. We also introduce bounds for Cartesian product graph Cm x Cn, m, n > 3.
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 d > 1. Let f be a graceful colouring for G. If f (u) = a, 1 < a < d, then there is a vertex v € N(u) with colour f (v) > d + a.
Proof. Let f (u) = a with 1 < a < d. If a = 1, 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 f (v) > 2a + (d — (a — 1) — 1) = d + a. □
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.
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.
Proof. 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.^
The following result was introduced by Byers (2018) in [3].
Lemma 2 (Byers in [3]). The graceful chromatic number of cycle Cn on n > 3 vertices is
Xg(Cn)={5: f n=5 (1.1)
Then, we will introduce some terminologies related with certain ladder graphs.
A ladder of 2m vertices, m > 2, denoted by Lm, is the Cartesian product graph of the path on m vertices and the path on two vertices. The ladder L2 is the cycle graph of four vertices. Assume that the vertices of Lm are vi,v2,..., vm,wi, w2,..., wm such that its edges are v^v^+i, wiwi+i : 1 < i < m — 1, vjwj : 1 < i < m. For m > 4, if the vertices vi and vm, and the vertices wi and wm are identified, then we obtain a prism Cm-i x P2. In this resulting Cm-i x P2, vi = vm, wi = wm, and edge viwi = vmwm. Due to this, we may call the ladder Lm as the open graph of Cm-i x P2 about the edge viwi.
On the other side, let Cm x P2,m > 3, be a prism. This prism has vertex set {vi, v2,..., vm, wi, w2,..., wm} and edge set
{vjvi+i, wjwj+i : 1 < i < m — 1} U {vivm, wiwm, } U {vjw» : 1 < i < m}.
After opening Cm x P2 about the edge viwi into the ladder Lm+i, the vertices vi and wi copy themselves into two copies each; the first copy of vi(resp. wi) is adjacent with v2(resp. w2), and the second copy of vi(resp. wi) is adjacent with vm(resp. wm). These last vertex copies in the ladder Lm+i are named as vm+i and wm+i, respectively. Therefore, if f a colouring for the prism Cm x P2, then in the ladder Lm+i we have f(vi) = f(vm+i as well as f(wi) = f(wm+i). In this case, we may also call Cm x P2 as the closed graph of Lm+i about the edges viwi and vmwm.
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.
Lemma 3. Using four different colours, the graph Cm x P2, with m > 3, m ^ 0 (mod 4), can not be gracefully coloured.
Proof. Let a, b, c and d be four different colours, and let m = 4k + r, 1 < r < 3. Consider the ladder Lm+i as the opened graph of Cm x P2. Let the vertex and edge sets of the ladder Lm+i be {vi,wi : 1 < i < m + 1} and {vivi+i, wiwi+i : 1 < i < m,viwi : 1 < i < m + 1}, respectively. Observe that the colour of vj (resp. wj) must be the same with the colour of wj+2 (resp. vj+2) or of wj-2 (resp. vj-2) for realizable integer j (realizable means in the range of discussion). Without loss of generality, let the colour of vi is a. Therefore, the colour of w4s+3 and of v4t+i 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 Cm x P2.
Case r = 1. If we take t = k, then we have f(vi) = a = f(v4k+i) = f(vm). Note that vm+i = v4k+2 is adjacent with vm. Thus, f(vm+i) can not be a to maintain proper colouring property. But, in Cm x P2, vertices vi and vm+i are identical which insist f(vm+i) = f(vi) = a. This implies a contradiction. So, for r = 1 the graph Cm x P2 can not be gracefully coloured.
Case r = 2. Applying a similar argument, by assuming the colour of vi is a, we have that f (wm+i) = f (w4k+3) = f (vi) = a. In graph Cm x P2, vertices wi and wm+i are identical. On the other side, wi is adjacent with vi , 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 (wm) = f (w4k+3) = f (vi) = a. We know that wm+i in Cm x P2 is identified with wi, and therefore is adjacent with both wm and vi. This implies that the induced edge colours of viwi(= viwm+i) and wiwm are the same which then contradicts the gracefulness property.
In any case we have proven that Cm x P2, m ^ 0(mod 4), can not be gracefully coloured using only 4 colours. □
Figure 1. A graceful colouring of C8 x P2.
2. Results on prism and generalized prism graphs
In this section, we will be dealing with the graceful chromatic number of prism Cm x P2 first, m > 3, and then with the graceful chromatic number of generalized prism graphs Cm x Pn, m,n > 3. As for some consequences, we also derive some bounds for graceful chromatic number of graph Cm x Cn, m,n > 3, for some specific values of m and n.
Our main discussion will be separated into two subsections: For Cm x P2, m > 3 and for Cm x Pn, with m, n > 3.
2.1. Prism graph Cm x P2 for m > 3.
Theorem 1. If m = 0 (mod 4), then the graceful chromatic number of graph Cm x P2 is equal
to 5.
Proof. Note that the graph Cm x P2 contains subgraph C4. Based on Lemma 2, we may conclude that xg(Cm x P2) > 4. Since all vertices of Cm x P2 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 Cm x P2 gracefully. To confirm this, we will do by introducing the following graceful colouring technique for Cm x P2 using only labels 1,2,4, and 5.
Let the vertices of Cm x P2 is the set
{v1+i,v2+i ,V3+i,V4+i ,W1+i ,W2+i ,W3+i, W4+ : i = 4k, k = 0,1, 2,... ,m/4 - 1} and its edge set is
{v1vm,w1 Wm, VjVi+1 ,WjWi+1,ViWj : i = 1, 2
m
- 1 .
Define a colouring f for Cm x P2 as follows.
f (Vi) =
1, if i III 1
4, if i = 2
5, if i = 3
2, if i = 0
(mod 4), (mod 4), (mod 4), (mod 4),
f (Wi) =
5, if i = 1 (mod 4),
2, if i = 2 (mod 4),
1, if i = 3 (mod 4),
4, if i = 0 (mod 4).
(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 Cm we have
{If(u) - f(w)|, If(w) - f(v)|} = {1,3}.
Furthermore, we also have
{If (vi) - f (Wi)| : 1 < i < m} = {2, 4}.
Remember that each vertex u in Cm x P2 has degree 3; say xi, and x3 are the vertices adjacent to u. From the function f we can immediately conclude that the set
{If(u) - f(xi)|, If(u) - f(X2)|, |f(u) - f(xs)|}
is equal to {1,2,3} or to {1,3,4}. Thus, the function f satisfies the property to become graceful colouring for Cm x P2. Therefore, xg(Cm x P2) = 5. □
Theorem 2.
to 6.
If m ^ 0 (mod 4), then the graceful chromatic number of graph Cm x P2 is equal
Proof. 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 C4k x P2 which is coloured as in (2.1). Let the ladder L4k+1 be the open graph of C4k x P2 about v1w1. Since Cm x P2 contains subgraph C4, 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 C4k x P2 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 Cm x P2 is at least 6. To conclude that xg(Cm x P2) = 6, we will proceed by showing that a graceful colouring exist with maximum colour 6, as follows.
Case 1: m = 1 (mod 4). First, consider C5 xP2 with vertex set {a1, a2, a3, a4, a5, b1, 62, 63, 64,65} and with edge set {a1a5,6165, aiai+1,6i6i+1 : i = 1 < i < 4} U {ai6i : 1 < i < 5}. Now, we colour vertices using the following function f:
/ (ai) = <
f 1, if
4, if 3, if
5, if 2, if
= 1, = 2, = 3, = 4, = 5,
/ (bi) =
5, if i = 1,
2, if i = 2,
6, if i = 3,
1, if i = 4,
4, if i = 5.
The coloured C5 x P2 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 from the coloured C5 x P2 above about a161. In the colours of and a6 are 1,2, 5,3,4, and 1, while the colours of 61,62,63,64,65, and 66 are
5, 4, 1, 6, 2, and 5.
Then, consider the open ladder L4k+1, for some positive integer k > 1, from the coloured C4k x P2 in Theorem 1 about v1w1. Here, the colours of v1 and w1 are also 1 and 5, respectively. The same colours are also for v4k+1 which is 1, and for w4k+1 which is 5. Based on (2.1), we have f (v4k) = 2, and f (w4k) = 4. By identifying v4k+1 with a6 and w4k+1 with 66, and maintaining the
i
Figure 2. A graceful colouring of C5 x P2.
other vertex colours, then we get a new ladder on 4(k + 1) + 2 vertices, L4(fc+1)+2, with graceful colouring.
Furthermore, we know that f (v2) = 4, f (w2) = 2, f (a2) = 2, and f (b2) = 4. Thus by identifying v1 with a1 and w1 with b1 in the ladder L4(fc+1)+2, we obtain C4(k+1)+1 x P2 with a graceful colouring.
From here, we may infer that the graceful chromatic number of the graph Cm x P2, for m = 1 (mod 4) is equal to 6.
Case 2: m = 2 (mod 4). First, consider C6 x P2 with vertex set
{a1, a2, a3, a4, a5, a6,61,62, &3, &4, &5, be},
and with edge set
{a1a6,6166,aiai+1,6ibi+1 : i = 1 < i < 5, a^bi : 1 < i < 6}. As a seed graph, we define the following colouring for C6 x P2 as follows.
f (ai) = <
By inspection we can verify that the above colouring for C6 x P2 is graceful. The diagram of the coloured graph is shown in Fig. 3.
Let the ladder of 7 vertices, L7, is the open graph from the C6 x P2 above about v1w1. We emphasize here that in this ladder L7, vertices a7 and b7 have colours 1 and 5, respectively; the same as the colours of a1 and b1 , respectively.
We use again the same ladder L4k+1, k > 1, as in Case 1. Now we identify v4k+1 with a7 and w4fc+1 with b7, and maintaining the other vertex colours. Then we get a new ladder on 4(k +1) + 3 vertices, L4(fc+1)+3, with graceful colouring.
Furthermore, we identify v1 with a1 and w1 with b1 in the ladder L4(fc+1)+3. Based on the previous colours, we know that the colours of v2,w2,a2,b2,v1 = a1,w1 = b1, are 4,2,3,6,1, 5, respectively. This means that after the last identification, the gracefulness colouring of C4(k+1)+2 are maintained. Thus, we may conclude that C4(-fc+1)+2 x P2 is with graceful colouring.
1, if i = 1, 5, if i = 1,
3, if i = 2, 6, if i = 2,
4, 1, if i if i = 3, = 4, f(bi) = 2, 5, if if i i = 3, = 4,
3, if i = 5, 6, if i = 5,
4, if i = 6, 2, V " if i = 6.
Figure 3. A graceful colouring of C6 x P2.
Figure 4. A graceful colouring of C10 x P2.
A graceful labeled C10 x P2 which is constructed using this method is depicted in Fig. 4.
From here, we may infer that the graceful chromatic number of the graph Cm x P2, 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 Cm x P2 with m = 3 (mod 4). We start with C3 x P2 with vertex set {a1,a2, a3,b1,b2,b3} and edge set {a3a1,a1a2,a2a3,b3b1,b1b2,b2b3,a1b1,a2b2,a3b3}. Then we colour C3 x P2 using the following colouring f.
( 1, if i = 1, ( 5, if i = 1,
f (ai) = < 3, if i = 2, f (bi) ^ 6, if i = 2, [4, if i = 3, [2, if i = 3.
We can immediately check that this colouring f is graceful. The diagram of the gracefully coloured graph C3 x P2 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 C3 x P2 is graceful. As we did for Case 1 and Case 2, first we will observe the open ladder L4 from C3 x P2 about aibi. In this L4, the
i
Figure 5. A graceful colouring of C3 x P2.
Figure 6. A graceful colouring of C7 x P2.
colour of vertices a4 = o1 = 1 and 64 = 61 = 5. Observe back the open ladder L4k+1 in Case 1 (and Case 2).
Now we identify v4k+1 with a4 and w4k+1 with 64 to obtain a graceful colouring ladder L4k+4. Let us denote the colouring as a. We can easily see that in this ladder we have a(o1) = a(v1) = 1 and a(61) = a(w1) = 5. Moreover, we have also a(a2) = f (a2) = 3, a(62) = f (62) = 6, a(v2) = 4, and a(w2) = 2. Thus, by identifying v1 with a1 and w1 with 61, we get a graceful colouring C4k+3 x P2, with graceful chromatic number is 6. See the labeled graph C7 x P2 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 Cm x P2, with m = 3 (mod 4) is also 6.
Since in all cases of m we proved that Cm x P2 has graceful chromatic number 6, we may conclude that xg(Cm x P2) = 6. □
2.2. Results on generalized prism graphs Cm x Pn, m, n > 3.
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 P3 x P3 can not be coloured by only four different colours. This observation gives
Xg(P3 x P3) > 5.
But, if we use only five colours 1,2,3,4 and 5, the center vertex of P3 x P3 must be 1 or 5. Then, by inspection we can show that using only five colours, we can not colour P3 x P3 gracefully. This gives the following lemma.
Lemma 4. The graceful chromatic number of the graph P3 x P3, \g (P3 x P3) > 6.
The following Lemma 5 will be an important tool for the proofs of our main results encountered in this section.
Lemma 5. The graceful chromatic number of the graph P5 x P5, xg (P x P5) > 7.
Proof. Let the vertices of P5 x P5 be V(P5 x P5) = {v^ : i,j = 0,1,2,3,4} and E(P5 x P5) = {vijvi(j+i),vij-v(i+1)j : i,j = 0,1,2,3}. Now, observe the subgraph P3 x P3 with V(P3 x P3) = {vij : i, j = 1, 2, 3} and
E(P3 x P3) = {vijv(i+i)j,vijv(i)(j+i) : i,j = 1, 2}.
In P5 x P5, every vertex of the subgraph P3 x P3 has degree 4. Based on Lemma 4, for gracefully colouring P3 x P3, we need at least five colours. If the colour 3 or 4 is assigned for a vertex of P3 x P3, then based on Lemma 1 the colour greater than or equal to 4 + 3 = 7 must appear in P5 x P5. If the colors 3 and 4 both are not assigned for any vertex of P3 x P3, then, since we need at least five colours, we need some color greater than or equal to 7 for gracefully colouring P5 x P5.D
Now, observe the graph P4 x P3. We will make use of this observation for facilitating the result which will be formulated in Lemma 6. Let V(P4 x P3) = {vj : i = 0,1, 2, 3; j = 0,1, 2}, and E(P4 x P3) = {vijVi(j+i) : i = 0,1, 2, 3; j =0,1} U {vjV(i+i),- : i = 0,1, 2; j = 0,1, 2}. The picture in Fig. 7 is the diagram of graph P4 x P3 with vertex names.
Figure 7. The graph P4 x P3 with vertex names.
In here, we will restrict a vertex colouring a for P4 x P3 as a(v0j) = a(v3j), Vj = 0,1,2. We will show that under this restriction, using only six colours, the vertex colouring a can not be graceful.
Let the six colours be 1,2,3,4, 5 and 6. Based on Lemma 1, since the degree of vertices v11 and v21 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 v11 and v21. In total, there are six different combinations for colouring these two vertices: {a(v11), a(v21)} = {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 a is not graceful. But, for the space consideration, we will only describe the detail process for combination {a(v11 ),a(v21)} = {1,2} as in Fig. 8. Note that the case a(v11) = a and a(v21) = b is similar to the case a(v11) = b and a(v21) = a.
The explanation of the colouring process in Fig. 8 is the following:
1) The colours a(v11) = 1 and a(v21) = 2 are fixed as the initial combination.
2) The next vertex colouring follows the following vertices order: v2o,v1o,voo,vo1 ,vo2,v12,v22. Note that a(v3j) := a(voj), Vj = 0,1,2, based on the restriction imposed for a.
3) For some colours x,y and z, a notation x/y/z means that we assign the colour ^(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 a(v1j) = a(v4j), j = 0,1,2, using exactly six different colours, we can not colour the graph P4 x P3 gracefully.
Figure 8. The colouring process for P4 x P3 with a(vii) = 1 and a(v21) = 2.
If we extend this last observation to graph P4 x Pn, n > 3, with
V (P4 x Pn)={vij : i=0,1, 2, 3; j=0,1,...,n - 1},
and
E(Pm x Pn) = {vijVi(j+1) : i=0,1, 2, 3; j=0,-2}u{vjV(m)j : i = 0,1, 2; j = 0,-1},
under restriction that a(voj) = a(v3j), j = 0,1,..., n — 1, we may also conclude that we need at least seven colours to maintain the colouring a becomes graceful for P4 x Pn. From this last observation we can formulate the following result.
Lemma 6. For n > 3, the graceful chromatic number of the graph C3 x Pn, xg(C3 x Pn) > 7.
Proof. The generalized prism graph C3 x Pn, n > 3, can be obtained by identifying vertices voj and v3j for every j = 0,1,2,..., n — 1 as it is in the last observation. By considering a graceful colouring a for the graph P4 x Pn under the above mentioned restriction, we are done. □
For facilitating the discussion of our main results in this section, we need the following definition, as we defined a ladder as an open graph of Cm x P2 in the previous section. Here we will define a similarone as an open graph from the graph Cm x Pn, m, n > 3. Let the vertex set of graph Cm x Pn, m, n > 3, be
{vij, 0 < i < m - 1, 0 < j < n - 1},
and its edge set be
{vijvkl, if i = k and |j — 1| = 1 or j = l and |i — k| = 1 (mod m)}.
Consider the open graph of Cm x Pn, m,n > 3, about the path P which has end vertices v00 and v0n, and has vertex set and edge set {v0j, j = 0,1,... ,n — 1} and {v0jv0(j+1), j = 0,1,... ,n — 2}, respectively. Denote this open graph by Lm+1,n. This graph is a grid graph having (m + 1) x n vertices which involves two copies of path P. These two copies of path P, each has vertices v0j, j = 0,1,..., n—1 and edges v0jv0(j+1), j = 0,1,..., n—2. In the open graph Lm+1,n, the vertices and edges of the second copy of P will be denoted by vmj, j = 0,1,..., n — 1, and vmjv(m)j+1), j = 0,1,..., n — 1, respectively. It is clear that the vertex vmj is adjacent with v(m-1)j for every j = 0,1,..., n — 2. In this case, Cm x Pn can be reconstructed from Lm+1,n by identifying vertex v0j and vmj for every j = 0,1,..., n — 1.
Theorem 3. For any positive integers m,n > 3, with m = 0 (mod 3), xg(Cm x Pn) = 7.
Proof. From Lemma 4 we know that the graceful chromatic number of Cm x Pn is at least seven. Now we will show that a graceful colouring exists for Cm x Pn such that it uses only seven different colours, and therefore xg(Cm x Pn) = 7.
Let the vertex set of Cm x Pn is {vij|0 < i < m — 1; 0 < j < n — 1}, and edge set
{vijvrs|i = r and |s — j| = 1 (mod n) or j = s and |i — r| = 1 (mod m)}. To this end, here we define a colouring function f for Cm x Pn as follows.
/ ) =
' 1, if = 0 (mod 3), j = 0 (mod 6),
5, if = 0 (mod 3), j = 1 (mod 6),
6, if i = 0 (mod 3), j = 2 (mod 6),
2, if i = 0 (mod 3), j = 3 (mod 6),
3, if i = 0 (mod 3), j = 4 (mod 6),
7, if i = 0 (mod 3), j = 5 (mod 6),
3, if i = 1 (mod 3), j = 0 (mod 6),
7, if i = 1 (mod 3), j = 1 (mod 6),
1, if i = 1 (mod 3), j = 2 (mod 6),
\ 5, if i = 1 (mod 3), j = 3 (mod 6),
6, if i = 1 (mod 3), j = 4 (mod 6),
2, if i = 1 (mod 3), j = 5 (mod 6),
6, if i = 2 (mod 3), j = 0 (mod 6),
2, if i = 2 (mod 3), j = 1 (mod 6),
3, if i = 2 (mod 3), j = 2 (mod 6),
7, if i = 2 (mod 3), j = 3 (mod 6),
1, if i = 2 (mod 3), j = 4 (mod 6),
5, if i = 2 (mod 3), j = 5 (mod 6).
(2.2)
An example of a graceful coloured graph C6 x Pn using (2.2) is shown in Fig. 9. In this figure we may also see the related open graph L7,n of C6 x Pn.
Figure 9. A graceful colouring of C6 x Pn, n > 3.
Fig. 9 also helps us to be able to check by inspection that f is a graceful colouring for the graph Cm x Pn, with m = 0 (mod 3). Therefore, we may conclude that this graph has chromatic number 7. □
Furthermore, based on (2.2) we see that for every i, 0 < i < m — 1, we have f(vj) = f(vik) provided |j — k| = 0 (mod 6).
Corollary 2. For any positive integers m,n > 3, with m = 0 (mod 3) and with n = 0 (mod 6),
Xg (Cm x Cn) = 7-
Proof. The proof of this corollary may be derived from (2.2). From Theorem 3 we conclude that xg(Cm x Pn) = 7, if m = 0 (mod 3), and n > 3. From (2.2) we know that f(vj) = f(vik) whenever |j — k| = 0 (mod 6). Thus, if n = 0 (mod 6), then if we identify vertex vi0 and vin for every i, 0 < i < m — 1 in Cm x Pn, then we get a graceful coloured graph Cm x Cn, m = 0 (mod 3) and n = 0 (mod 6). Therefore, we may conclude that xg (Cm x Cn) = 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 Cm x Pn, 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 < Xg (Cm x Pn) < 8.
Proof. We will make use of prism graph C4 x Pn as the seed of our graceful colouring construction. We first introduce a colouring for the graph C4 x Pn, n > 3.
Figure 10. A graceful colouring of C4 x Pn.
Let the vertex set of C4 x Pn is
{vij|0 < i < 3; 0 < j < n — 1},
and edge set
{vijvrs| i = r and |s — j| = 1 (mod n) or j = s and |i — r| = 1 (mod 4)}. To this end, we define a colouring function f as follows.
/(v ij) = <
' 1, if = 0 (mod 4), j = 0 (mod 4),
2, if i = 0 (mod 4), j = 1 (mod 4),
6, if i = 0 (mod 4), j = 2 (mod 4),
5, if i = 0 (mod 4), j = 3 (mod 4),
3, if i = 1 (mod 4), j = 0 (mod 4),
4, if i = 1 (mod 4), j = 1 (mod 4),
8, if i = 1 (mod 4), j = 2 (mod 4),
7, if i = 1 (mod 4), j = 3 (mod 4),
6, if i = 2 (mod 4), j = 0 (mod 4),
7, if i = 2 (mod 4), j = 1 (mod 4),
3, if i = 2 (mod 4), j = 2 (mod 4),
2, if i = 2 (mod 4), j = 3 (mod 4),
4, if i = 3 (mod 4), j = 0 (mod 4),
5, if i = 3 (mod 4), j = 1 (mod 4),
1, if i = 3 (mod 4), j = 2 (mod 4),
8, if i = 3 (mod 4), j = 3 (mod 4).
(2.3)
For an illustration one can see in Fig. 10
Fig. 10 helps us to see that (2.3) gives a graceful colouring for C4 x Pn for every n > 3 with Xg(C4 x Pn) < 8. Therefore, based on Lemma 4, we may conclude that 7 < xg(C4 x Pn) < 8.
Furthermore, the graceful colouring of Cm x Pn, with m = 1 (mod 3) and n > 3 in general, is obtained by extending graceful coloured graph C4 x Pn using the prism graph C3 x Pn which has colouring as we will show below. Let the vertex set of C3 x Pn is
{vj| 0 < i < 2; 0 < j < n - 1}
Figure 11. A graceful colouring of C3 x Pn.
and edge set
{vijvrs| i = r and |s — j| = 1 (mod n) orj = s and |i — r| = 1 (mod 3)}. To this end, we define a colouring function f as follows.
f (vij) =
1, if = 0 (mod 3), j = 0 (mod 4),
2, if i = 0 (mod 3), j = 1 (mod 4),
6, if i = 0 (mod 3), j = 2 (mod 4),
5, if i = 0 (mod 3), j = 3 (mod 4),
3, if i = 1 (mod 3), j = 0 (mod 4),
4, if i = 1 (mod 3), j = 1 (mod 4),
\ 8, if i = 1 (mod 3), j = 2 (mod 4),
7, if i = 1 (mod 3), j = 3 (mod 4),
6, if i = 2 (mod 3), j = 0 (mod 4),
7, if i = 2 (mod 3), j = 1 (mod 4),
3, if i = 2 (mod 3), j = 2 (mod 4),
2, if i = 2 (mod 3), j = 3 (mod 4).
(2.4)
The diagram of coloured graph L4,n from C3 x Pn is depicted in Fig. 11. The coloured graph C3 x Pn is obtained by identifying v0j and v3j for all j, 0 < j < n — 1. We can immediately observe that (2.4) gives a graceful colouring for the prism graph C3 x Pn with xg(C3 x Pn) < 8. Again based on Lemma 4, we conclude that 7 < xg(C3 x Pn) < 8.
For producing a graceful colouring for Cm x Pn, m = 1 (mod 3) we use L5,n from C4 x Pn and L4,n from C3 x Pn, by identifying v5j of L5,n and v0j of L4,n for all j, 0 < j < n — 1. This identification results in a graceful coloured grid graph Ls,ra. Then, if we identify v8j of L8,ra and v0j of L4,n for all j, 0 < j < n — 1, we get a graceful coloured grid graph L11,n. Continuing the same procedure, then we get a graceful coloured grid graph L(m+1),n. Then by identifying vertex v0j and vmj from L(m+1),n we obtain Cm x Pn with m = 1 (mod 3) and n > 3. □
As one consequence, as we formulated Corollary 2 based on Theorem 3, we also formulate a corollary based on Theorem 4 as the following.
Corollary 3. If m = 1 (mod 3) and n = 0 (mod 4), then 7 < xg(Cm x Cn) < 8.
Now we go to the next case m = 2 (mod 3). The result is formulated in the following theorem.
Figure 12. A graceful colouring of grid graph L6,n of C5 x Pn.
Theorem 5. If m = 2 (mod 3), then 7 < Xg(Cm x Pn) < 8.
Proof. To proof this theorem, we will start with a graceful colouring for C5 x Pn, m = 2 (mod 3), n > 3. We introduce the following colouring for the graph C5 x Pn, n > 3.
/ (vj) =
' 1, if = 0 (mod 5), j = 0 (mod 4),
2, if = 0 (mod 5), j = 1 (mod 4),
6, if = 0 (mod 5), j = 2 (mod 4),
5, if i = 0 (mod 5), j = 3 (mod 4),
3, if i = 1 (mod 5), j = 0 (mod 4),
4, if i = 1 (mod 5), j = 1 (mod 4),
8, if i = 1 (mod 5), j = 2 (mod 4),
7, if i = 1 (mod 5), j = 3 (mod 4),
6, if i = 2 (mod 5), j = 0 (mod 4),
7, if i = 2 (mod 5), j = 1 (mod 4),
3, if i = 2 (mod 5), j = 2 (mod 4),
2, if i = 2 (mod 5), j = 3 (mod 4),
4, if i = 3 (mod 5), j = 0 (mod 4),
5, if i = 3 (mod 5), j = 1 (mod 4),
1, if i = 3 (mod 5), j = 2 (mod 4),
8, if i = 3 (mod 5), j = 3 (mod 4),
7, if i = 4 (mod 5), j = 0 (mod 4),
8, if i = 4 (mod 5), j = 1 (mod 4),
4, if i = 4 (mod 5), j = 2 (mod 4),
3, if i = 4 (mod 5), j = 3 (mod 4).
(2.5)
The diagram for coloured grid graph L6,n of C5 x Pn, which is derived from (2.5), can be seen in Fig. 12. Using this diagram we may conclude that the colouring is graceful. It is clear that Xg(C5 x Pn) < 8.
The process of expanding to get coloured graph Cm x Pn, m = 2 (mod 3), n > 3, is similar to the previous process as was described in the proof of Theorem 4. Here we use graceful coloured grid graph L6,n from graceful coloured graph C5 x Pn, and graceful coloured grid graph L4,n from graceful coloured graph C3 x Pn. Again by considering Lemma 4,we then conclude that
7 < Xg(C5 x Pra) < 8. □
Similar to the previous corollaries, here we formulate the following corollary as a consequence of Theorem 5.
Corollary 4. If m = 2 (mod 3) and n = 0 (mod 4), then 7 < xg(Cm x Cn) < 8.
3. Conclusion
In the discussion above, it has been proven that prism graph Cm x P2 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 Cm x Pn 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 < xg(Cm x Pn) < 8. Based on these results, we could also derive some exact and bound values of graceful chromatics number of Cm x Cn for certain m,n > 3. Regarding this last observation, we propose the following open problem and conjecture. Conjecture. If m ^ 0 (mod 3) and n > 3, then xg(Cm x Pn) = 8. Open problem. What is xg(Cm x Cn), if m ^ 0 (mod 3), m,n > 3?
Acknowledgements
The authors express their deep gratitude to referees for their invaluable corrections and suggestions. The first author thanks to the Research and Social Service Board of Universitas Pendidikan Ganesha for the provided support to conduct this research.
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