The locating-chromatic number of certain Halin graphs

The locating-chromatic number of certain Halin graphs Ira Apni Purwasih and Edy Tri Baskoro Citation: AIP Conf. Proc. 1450, 342 (2012); doi: 10.1063/1.4724165 View online: http://dx.doi.org/10.1063/1.4724165 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1450&Issue=1 Published by the American Institute of Physics.

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The Locating-Chromatic Number of Certain Halin Graphs Ira Apni Purwasih and Edy Tri Baskoro Combinatorial Mathematics Research Group Faculty of Mathematics and natural Sciences Institut Teknologi Bandung (ITB) Jalan Ganesa 10 Bandung 40132 Indonesia [email protected], [email protected] Abstract. The locating-chromatic number of a graph G can be defined as the cardinality of a minimum ordered partition Π of the vertex-set V (G) such that every two vertices in G have different coordinates with respect to Π and every two adjacent vertices in G are not in the same partition class in Π. In this case, the coordinate of a vertex v is defined as the distances from the vertex v to all partition classes in Π. In this paper, we discuss the locating-chromatic number of Halin graph, namely a planar graph H = T ]C which constructed from a plane embedding of a tree T with at least four vertices, by connecting all the leaves of T (the vertices of degree 1) to form a cycle C that passes around the tree in the natural cyclic order defined by the embedding of the tree. In particular, we investigate the locating-chromatic number of the Halin graph H = T ]C, where T is a subdivision of a star and a double star. Keywords: locating-chromatic number, Halin Graph PACS: 02.10.Ox

INTRODUCTION The concept of locating-chromatic number was introduced by Chartrand et. al. [8] in 2002, as a marriage of two major concepts in the graph theory, namely graph coloring and partition dimension. Let G = (V, E) be a finite, simple, and connected graph. The distance d(u, v) between vertices u and v in G is the length of a shortest path connecting u and v in G. Let c be a proper k-coloring of V (G) which induces an ordered partition Π = {R1 , R2 , · · · , Rk } of V (G), where Ri is the set of all vertices colored by i in G. The color code cΠ (v) of vertex v is the ordered k-tuple (d(v, R1 ), d(v, R2 ), ..., d(v, Rk )), where d(v, Ri ) = min{d(v, x)|x ∈ Ri } for 1 ≤ i ≤ k. In particular, if d(v,U) 6= d(w,U) for some set U then we shall say that u and v are distinguished by U or x and y are distinguishable. If every two vertices have different color codes then c is called a locating k-coloring of G. The locating-chromatic number of graph G, denoted by χL (G), is the smallest integer k such that G has a locating k-coloring. Problem of determining the locating-chromatic number of a graph is an NP-hard problem. This means that no efficient algorithm to determine the locating-chromatic number of an arbitrary graph. Therefore, many heuristics approaches have been developed to determine this graph parameter. On the other hand, people also study this problem by determining the locating-chromatic number of certain classes of graphs, for instances paths, cycles, certain trees and a few others. Characterization study of graphs with respect to this graph parameter has been also

carried out. However, the results are still little and not yet satisfactory. In [8], Chartrand et al. determined the locatingchromatic numbers of some well-known classes such as paths, cycles, complete multipartite graphs and double stars. For double star, they found that χL (Sa,b ) = b + 1 for integers a and b with 1 ≤ a ≤ b and b ≥ 2. They determine that χL (Cn ) = 3 if n is odd and χL (Cn ) = 4 if n is even. Furthermore, in [9] they showed that there always exists a tree of order n ≥ 5 having locating-chromatic number k if and only if k ∈ {3, 4, ..., n − 2, n}. In [3], Asmiati et al. derived the locating-chromatic number for some class of trees, especially a class of trees obtained as an amalgamation of n (not necessarily isomorphic) stars. Furthermore, Asmiati at al., in [1], investigated the locating-chromatic number of a firecracker graph, i.e. a tree constructed from n stars by connecting one leaf from each star to form a path Pn . For a given tree T with at least four vertices, a Halin graph H = T ] C is a planar graph which is constructed from a plane embedding of T , by connecting all the leaves (the vertices of degree 1) of T to form a cycle C that passes around the tree in the natural cyclic order defined by the embedding of the tree. The wheel Wn on n + 1 vertices is one of example of a Halin graph of a star Sn+1 . The following theorem was proved by Behtoei and Omoomi (2011) in [7]. Theorem A Let Wn be a wheel on n + 1 vertices and n0 = min{k ∈

The 5th International Conference on Research and Education in Mathematics AIP Conf. Proc. 1450, 342-345 (2012); doi: 10.1063/1.4724165 © 2012 American Institute of Physics 978-0-7354-1049-7/$30.00

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N | n = 12 (k3 − k2 )}. Then,   1 + χL (Cn ) if 3 ≤ n ≤ 9; 1 + n0 if n 6= 21 (n30 − n20 ) − 1 and n > 9 ; χL (Wn ) =  2 + n0 if n = 12 (n30 − n20 ) − 1 and n > 9. In this paper we will determine the locating-chromatic number of certain Halin graphs. In particular we investigate the Halin graphs H = T ] C, where T is a subdivision of a star or a double star.

for any i ∈ {2, 4, · · · , n − 1} then the color codes of x and y under c0 is also different. Therefore, c0 is locatingcoloring of Wn−1 with t − 1 colors, a contradiction. So, χL (H1 ) ≥ t − 1. Next, we will show that χL (H1 ) ≤ t. Let c be a locating t-coloring of Wn−1 with c(v) = 1. If n ≥ 5, without loss of generality, consider the Halin graph H1 = T1 ] C which is isomorphic to a subdivision of a wheel Wn−1 on one edge vu1 in k1 times such that c(v) = 1, c(u1 ) = a and c(u2 ) = c(un−1 ) = b. Note that if the selection of u1 is not possible then select any u1 . Now, construct a t-coloring c0 of H1 by defining

RESULTS For positive integers k1 , k2 , · · · , kr and r, let S[G(k1 , k2 , · · · , kr )] be a graph obtained by a subdivision of graph G on chosen e1 , e2 , · · · , er edges in k1 , k2 , · · · , kr times respectively. Let Sn be a star on n vertices and Sn,m a double star on n + m vertices. In this section, we calculate the locating-chromatic numbers of Halin graphs H = T ] C, where T = S[Sn (k1 )], S[S4 (k1 , k2 , k3 )] or S[S3,3 (k1 , k2 )]. Let T1 = S[Sn (k1 )] be a subdivision of a star Sn on one edge in k1 times. Let H1 = T1 ] C, where V (H1 ) = {v, u1 , u2 , · · · , un−1 } ∪ {w1 , w2 , · · · , wk1 }, and E(H1 ) = {vui |2 ≤ i ≤ n − 1} ∪ {u1 u2 , u2 u3 , · · · , un−1 u1 } ∪ {vw1 , w1 w2 , · · · , wk1 −1 wk1 , wk1 u1 }. In this case, the obtained Halin graph H1 = T1 ] C is isomorphic to a graph obtained by a subdivision of one edge vu1 of a wheel Wn−1 on k1 times. Then, we obtain the following theorem. Theorem 1. Let H1 = S[Sn (k1 )] ] C, for n ≥ 4, k1 ≥ 1. Then, χL (Wn−1 ) − 1 ≤ χL (H1 ) ≤ χL (Wn−1 ). Proof. Let χL (Wn−1 ) = t. Since n ≥ 4 then t ≥ 4. Firstly, we will show that χL (H1 ) ≥ t − 1. For a contradiction assume there exists a (t − 2)-locating c of H1 . Now, construct a new coloring c0 on Wn−1 by removing all vertices wi in H1 such that   t − 1, if w = u1 c(ui ), if w = ui for 2 ≤ i ≤ n − 1, c0 (w) =  c(v), if w = v. By the definition of c0 , we have that for any wi there exists u j such that c(wi ) = c(u j ). Since otherwise, by removing all colors of vertices wi , the coloring c0 on Wn−1 will have t − 1 colors, a contradiction with χL (Wn−1 ) = t. Now, let x and y be two distinct vertices of Wn−1 with c0 (x) = c0 (y). Then, x, y 6∈ {v, u1 }. So, the color codes of x and y, under the coloring c on H1 , must be different and they are distinguished by the vertices in Wn−1 . This is true since d(x, wi ) = d(y, wi ) ≥ 2 for each wi ∈ H1 and all colors of wi appear in the vertices ui . Since c0 (ui ) = c(ui )

 c(ui ),    c(v),    a, 0 c (w) =       1,

if w = ui for 2 ≤ i ≤ n − 1, if w = v, if (w = u1 and k1 is even), or (w = wi for odd i), if w = u1 and k1 is odd, or (w = wi for even i).

If χL (Wn−1 ) = 4 then the selection of u1 with above criteria is always possible. If χL (Wn−1 ) ≥ 5 then select any u1 . By the definition of c0 above, we will show that c0 is a locating coloring on H1 . Let x, y ∈ V (H1 ) with c0 (x) = c0 (y). Then, the color codes of any x = ui and y = wi are different since d(x, S) 6= d(y, S) for some color set S not containing v, u1 , u2 or un−1 . If n = 4 and k1 is even then define c0 (v) = 1, c0 (u1 ) = 2, c0 (u2 ) = 3, c0 (u3 ) = 4, and c0 (wi ) = 1 for even i and c0 (wi ) = 2 for odd i. If n = 4 and k1 is odd then define c0 (v) = c0 (u1 ) = 1, c0 (u2 ) = 3, c0 (u3 ) = c0 (uk1 ) = 4, and c0 (wi ) = 1 for even i and c0 (wi ) = 2 for odd i and i < k1 . It is easy to verify that c0 is locating coloring of H1 . Theorem 2. Let H1 = S[Sn (k1 )] ] C for n ≥ 4 and k1 6= 2. Let V (H1 ) = {v; u1 , u2 , · · · , un−1 , w1 , w2 , · · · , wk1 } with v, vu1 , ui , wi are the center vertex, subdivision edge, cycle and subdivision vertices, respectively. Let Π be an ordered partition of V (Wn−1 ) induced by a minimum locating coloring c. Then, χL (H1 ) = χL (Wn−1 ) − 1 if χL (Wn−1 ) ≥ 5 and one of the following is satisfied: i. Vertex u1 belongs to a singleton color class in Π and c(u2 ) = c(un−1 ). ii. Vertex u1 belongs to a singleton color class in Π, c(u2 ) 6= c(un−1 ), and u2 (or un−1 ) with any other vertex is not only distinguished by u1 . Proof. Let Π be an ordered partition of V (Wn−1 ) by a minimum locating t-coloring c. Assume that one of the two conditions above is satisfied. Note that if the first condition is satisfied then c(u3 ) 6= c(un−2 ). Now, construct a new coloring c0 on H1 satisfying:

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Case 1. k1 is odd.   c(ui ), if w = ui for 2 ≤ i ≤ n − 1, c(v), if w = v, u1 or w = wi for even i c0 (w) =  c(u ), if w = w for odd i. i 2 Case 2. k1 is even.  c(ui ),   c(v), 0 c (w) =   c(u2 ), c(u3 ),

connecting y to z goes will pass through a. By Lemma 2 both paths must be odd length. However, in this case, the length of the clear path connecting x to z will be even, a contradiction.

if w = ui for 2 ≤ i ≤ n − 1, if w = v, u1 or w = wi for even i 6= k1 , if w = wi for odd i, if w = wk1 .

Thus, c0 has only t − 1 distinct colors. Under the new coloring c0 , all the color codes of the vertices in Wn−1 are the same except the color codes of u1 , u2 , un−1 . However, each of them is distinguishable with other vertices in Wn−1 . Now, consider the subdivision vertices of H1 . each of them is the same class with either v or u2 . It is easy to verify that they have different color codes. Therefore, c0 is a locating (t − 1)-coloring on H1 . By Theorem 1, χL (H1 ) = t − 1. From Theorem A, we know that χL (W6 ) = 5 and χL (W8 ) = 5. Let c be a minimum locating-coloring on W6 and W8 such that c(v, u1 , u2 , u3 , u4 , u5 , u6 ) = (1, 5, 4, 2, 4, 3, 2) and c(v, u1 , u2 , u3 , u4 , u5 , u6 , u7 , u8 ) = (1, 5, 4, 2, 3, 2, 3, 4, 2), respectively. Note that H10 = S[S7 (k1 )] ] C and H100 = S[S9 (k1 )] ] C, satisfy the conditions of Theorem 2 (ii) by considering the coloring c on W6 and W8 , respectively. Now, consider any graph G with χL (G) = 3. Let Π = {R1 , R2 , R3 } be an ordered partition of V (G) induced by a minimum locating coloring c on G. A vertex v ∈ G is called a dominant vertex if d(v, Ri ) = 0 for v ∈ Ri and 1 for otherwise. A path connecting two dominant vertices in G is called a clear path if all of its internal vertices are not dominant. Then, we have the following lemmas. Lemma 1. [2] Let G be a graph with χL (G) = k. Then, there are at most k dominant vertices in G and all of them must receive different colors. Lemma 2. [2] or [1] Let G be a graph with χL (G) = 3. Then, the length of any clear path in G is odd. Furthermore, we have the following lemma. Lemma 3. Let G be a graph with χL (G) = 3. Then, all dominant vertices must be contained in one path. Proof. If G contains two dominant vertices then it is clear that they are contained in a path. Now, assume that G has three dominant vertices, x, y and z. For a contradiction assume that they are not in some path in G. Then, there exists vertex a such that a clear path connecting x to y will go through a and a clear path

Lemma 4. Let G be a graph containing two intersecting cycles. If all common edges of the two intersecting cycles induce a path with even length, and the length of two cycles have different parities then χL (G) ≥ 4. Proof. Of course, χL (G) ≥ 2. For a contradiction, assume that χL (G) = 3. Let P0 be the induced path by all common edges of the two intersecting cycles C1 and C2 in G. Let x and y be the end-vertices of P0 . For i = 1, 2, let Pi be the path from x to y induced by all edges of Ci . Then, the length of paths P1 and P2 have different parities. Since G contains an odd cycle then G must have exactly three dominant vertices. Let a, b, c be the dominant vertices of G. By Lemma 3, these dominant vertices must be contained in a path P from a to c passing b. Without loss of generality, there are three cases to be considered. Case 1. All a, b, c ∈ P0 . Then, P is a subpath of P0 . By Lemma 2, two subpaths of P, namely one from a to b, and one from b to c, have odd length. Since the length of P0 is even, then the subpath of P0 from x to a and the one from c to y have the same parity. Therefore, there will be a clear path of even length from c to a passing through all edges of either P1 or P2 , a contradiction. Case 2. a, b ∈ P0 and c ∈ V (P1 ). Let P be a path from a to c passing vertices b, y and some vertices of P1 only. Now consider two clear paths, namely the path Q1 from a to b passing x, all edges of P2 and y, and the path Q2 from a to c passing x, all edges of P2 and y. By Lemma 2 both paths have odd length. Therefore, the subpath of Q1 from y to b and the subpath of Q2 from y to c have the same parity. Thus, the clear path from b to c passing y and some edges of P1 only has even length, a contradiction. Case 3. a ∈ P1 , b ∈ P0 and c ∈ P2 . Consider the clear path Q1 from a to b passing through y. By Lemma 2 the lengths of the subpath of Q1 from a to y and the subpath of Q1 from y to b have different parities. Without loss of generality, the length of the subpath Q1 from a to y is even and the subpath of Q1 from y to b is odd. Thus, the subpath of P2 from y to c is also even, by Lemma 2. Since the length of P0 is even then The subpath of P0 from b to x is odd. Therefore the lengths of subpath of P1 from x to a and subpath of P2 from x to c are even. Therefore the lengths of C1 and C2 has the same parity, a contradiction. Therefore, in any case we conclude that χL (G) ≥ 4.

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Let T2 = S[S4 (k1 , k2 , k3 )] be a graph obtained by subdivision of a star S4 on all edges in k1 , k2 , k3 times, respectively. Let H2 = T2 ] C, where V (H2 ) = {v, u1 , u2 , u3 } ∪ {x1 , x2 , · · · , xk1 } ∪ {y1 , y2 , · · · , yk2 } ∪ {z1 , z2 , · · · , zk1 } and E(H2 ) = {vx1 , x1 x2 , · · · , xk1 −1 xk1 , xk1 u1 } ∪ {vy1 , y1 y2 , · · · , yk2 −1 yk2 , yk2 u2 } ∪ {vz1 , z1 z2 , · · · , zk3 −1 zk3 , zk3 u3 } ∪ {u1 u2 , u2 u3 , u3 u1 }. In this case, the obtained Halin graph H2 = T2 ] C is isomorphic to a graph obtained by a subdivision of three spoke edges of a wheel W3 on k1 , k2 , k3 times, respectively. Then, we obtain the following theorem. Theorem 3. Let H2 = S[S4 (k1 , k2 , k3 )] ]C, for any positive integers k1 , k2 and k3 . Then χL (H2 ) = 4. Proof. First, we will show that χL (H2 ) ≥ 4. Two of {k1 , k2 , k3 } must be the same parity. Assume, without lost of generality, k1 and k2 have the same parity. Then, H2 contains two intersecting cycles with the common edges form a path P0 from u1 to u2 through v. These two cycles are (u1 , · · · , v, · · · , u2 , u1 ) and (u1 , · · · , v, · · · , u2 , u3 , u1 ). The lengths of these two cycles have different parities. Therefore, by Lemma 4, χL (H2 ) ≥ 4. Now, consider a 4-coloring c of H2 such that:  1, if (w = xi for even i and i 6= k1 ), or      (w = u1 , v, yi , or zi for even i),  c(w) = 2, if w = xk1 , zi or yi for odd i,    3, if w = u2 or w = xi for odd i,    4, if w = u3 . It is easy to see that the color codes of all vertices are distinct. Therefore, χL (H2 ) = 4. For m, n ≥ 3, a double star Sm,n is a tree having exactly one vertex of degree m, one vertex of degree n, and the remaining vertices of degree one. Let S[S3,3 ] be the graph derived from a subdivision two pendant edges attached to one vertex of degree 3 in k1 and k2 , respectively. Next, we determine the locating-chromatic number of the Halin graph H3 = S[S3,3 (k1 , k2 )] ] C. Let V (H3 ) = {u, u1 , u2 , v, v1 , v2 } ∪ {s1 , s2 , ..., sk1 } ∪ {t1 ,t2 , ...,tk2 } and E(H3 ) = {uv, uu1 , uu2 , vv1 , vv2 } ∪ {vs1 , s1 s2 , ..., sk1 −1 sk1 , sk1 v1 } ∪ {vt1 ,t1t2 , ...,tk2 −1tk2 ,tk2 v2 } ∪{u1 u2 , u2 v2 , v2 v1 , v1 u1 }.

Now, consider a 4-coloring c of H3 such that:  1, if w = u1 or w = si for odd i,    2, if w = u, v , or w = t for odd i, i 1 c(w) =  3, if w = v , 2    4, otherwise. It is easy to see that the color codes of all vertices of H3 are distinct. Therefore, χL (H3 ) = 4.

ACKNOWLEDGMENTS This work was supported by the Directorate General of Higher Education, Ministry of National Education Indonesia, Research Grant: 041/K01.1/IMHERE ITB/SPK/2011.

REFERENCES 1. Asmiati, E.T. Baskoro, H. Assiyatun, D. Suprijanto, R. Simanjuntak, S. Uttunggadewa, The Locating-Chromatic Number of Firecracker Graphs, Far East Journal of mathematical Sciences, 63:1 (2012), 11-23. 2. Asmiati, E.T. Baskoro, Characterizing all graphs containing cycles with locating-chromatic number 3, to appear in AIP Proceedings, 2012. 3. Asmiati, H. Assiyatun, E.T. Baskoro, Locating-chromatic number of Amalgamation of stars, ITB J. Sci., 43 A (1) (2011) 1-8. 4. E. T. Baskoro and I. A. Purwasih. The locating-chromatic number for corona product of graphs, to appear in East West Journal of Mathematics. 5. A. Behtoei, B. Omoomi. On the locating chromatic of Kneser graphs, Discrete App. Math., 159 (2011), 2214-2221. 6. A. Behtoei, B. Omoomi. On the locating chromatic of cartesian product of graphs, to appear in Ars Combin (2011). 7. A. Behtoei, B. Omoomi. The locating-chromatic number of the join of graphs, to appear in Discrete App. Math. (2011). 8. G. Chartrand, D. Erwin, M.A. Henning, P.J. Slater, P. Zhang, The locating-chromatics number of a graph, Bull. Inst. Combin. Appl, 36 (2002), 89-101. 9. G. Chartrand, D. Erwin, M.A. Henning, P.J. Slater, P. Zhang, Graph of order n with locating-chromatic number n-1, Discrete Math., 269 (2003), No.1-3, 65-79.

Theorem 4. Let H3 = S[S3,3 (k1 , k2 )] ]C, for any positive integer k1 and k2 . Then χL (H3 ) = 4. Proof. If k1 and k2 have the same parity then the path P0 = (v2 ,tk2 · · · ,t1 , v, s1 , · · · , sk1 , v1 ) has an even length. The path P0 is the intersection of two cycles C1 = P0 ∪ (v1 v2 ) and C2 = P0 ∪ (v1 u1 , u1 u, uu2 , u2 v2 ). The lengths of these two cycles have different parities. Then, by Lemma 4, χL (H3 ) ≥ 4.

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