Punjab University Journal of Mathematics (ISSN 1016-2526) Vol. 49(3)(2017) pp. 119-132 Some Families of Convex Polytopes Labeled by 3-Total Edge Product Cordial Labeling Umer Ali Department of Mathematics, UMT Lahore, Pakistan. Email: [email protected] Muhammad bilal Department of Mathematics, UMT Lahore, Pakistan. Email: [email protected] Sohail Zafar Department of Mathematics, UMT Lahore, Pakistan. Email: [email protected] Zohaib Zahid Department of Mathematics, UMT Lahore, Pakistan. Email: zohaib [email protected] Received: 03 January, 2017 / Accepted: 10 April, 2017 / Published online: 18 August, 2017 Abstract. For a graph G = (VG;EG), consider a mapping h : EG ! ∗ f0; 1; 2; : : : ; k − 1g, 2 ≤ k ≤ jEGj which induces a mapping h : VG ! ∗ Qn f0; 1; 2; : : : ; k − 1g such that h (v) = i=1 h(ei)( mod k), where ei is an edge incident to v. Then h is called k-total edge product cordial ( k- TEPC) labeling of G if js(i) − s(j)j ≤ 1 for all i; j 2 f1; 2; : : : ; k − 1g: Here s(i) is the sum of all vertices and edges labeled by i. In this paper, we study k-TEPC labeling for some families of convex polytopes for k = 3. AMS (MOS) Subject Classification Codes: 05C07 Key Words: 3-TEPC labeling, The graphs of convex polytopes. 119 120 Umer Ali, Muhammad Bilal, Sohail Zafar and Zohaib Zahid 1. INTRODUCTION AND PRELIMINARIES Let G be an undirected, simple and finite graph with vertex-set VG and edge-set EG. Order of a graph G is the number of vertices and size of a graph G is the number of edges. Graph labeling is a map that assigns integers to VG or EG or both subject to particular condition(s). In graph G a mapping from VG (or EG) to positive integers is called vertex (or edge) labeling and a mapping from VG [ EG to positive integers is known as total labeling. In order to understand cordial labeling h and its types, we need the following notations: (1) vh(i) is the number of vertices labeled by i; (2) eh(i) is the number of edges labeled by i; (3) vh(i; j) = vh(i) − vh(j); (4) eh(i; j) = eh(i) − eh(j) and (5) the sum of all vertices and edges labeled by i is s(i) i.e. s(i) = vh(i) + eh(i). Cahit introduced cordial labeling in [9]. ∗ Definition 1.1. Let h : VG −! f0; 1g be a mapping that induces h : EG −! f0; 1g such that h∗(uv) = jh(u) − h(v)j for each edge uv, then h is called cordial labeling if it satisfies jvh(1; 0)j ≤ 1 and jeh(1; 0)j ≤ 1. Product cordial labeling was introduced by Sundaram et al. in [15]. ∗ Definition 1.2. Let h : VG ! f0; 1g be a mapping that induces h : EG ! f0; 1g such that h∗(uv) = h(u)h(v) for each edge uv, then h is called product cordial labeling if it satisfies jvh(1; 0)j ≤ 1 and jeh(1; 0)j ≤ 1. In 2006, the total product cordial labeling was developed by Sundaram et al. in [16]. ∗ Definition 1.3. Let h : VG ! f0; 1g be a mapping that induces h : EG ! f0; 1g such that h∗(uv) = h(u)h(v) for each edge uv, then h is called total product cordial labeling if it satisfy js(0) − s(1)j ≤ 1. Definition 1.4. [13] Let h : VG ! f0; 1; : : : ; k − 1g, 2 ≤ k ≤ jEGj be a mapping that ∗ ∗ induces h : EG ! f0; 1; : : : ; k − 1g such that h (uv) = h(u)h(v)(mod k) for each edge uv, then h is called k-total product cordial labeling if it satisfy js(a) − s(b)j ≤ 1 for all a; b 2 f0; 1; : : : ; k − 1g. In 2012, Vaidya and Barasara introduced edge product cordial labeling (see [17]). ∗ Definition 1.5. Let h : EG ! f0; 1g be a mapping that induces h : VG ! f0; 1g such ∗ that h (u) = h(e1)h(e2) : : : h(en) for edges e1; e2; : : : ; en incident to u, then h is called edge product cordial labeling if it satisfies jvh(0; 1)j ≤ 1 and jeh(0; 1)j ≤ 1. Definition 1.6. Let h : EG −! f0; 1; : : : ; k − 1g, 2 ≤ k ≤ jEGj be a mapping that ∗ ∗ induces h : VG ! f0; 1; : : : ; k − 1g such that h (u) = h(e1)h(e2) : : : h(en)(mod k) for edges e1; e2; : : : ; en incident to u, then h is called k-edge product cordial labeling if it satisfies jvh(a; b)j ≤ 1 and jeh(a; b)j ≤ 1 for 0 ≤ a < b ≤ k − 1. ∗ Definition 1.7. [18] Let h : EG ! f0; 1g be a mapping that induces h : VG ! f0; 1g ∗ such that h (u) = h(e1)h(e2) : : : h(en) for edges e1; e2; : : : ; en incident to u, then h is called a total edge product cordial labeling if it satisfy js(0) − s(1))j ≤ 1. Some Families of Convex Polytopes Labeled by 3-Total Edge Product Cordial Labeling 121 In 2015, Azaizeh et al. provide the k-total edge product cordial labeling in [5]. Definition 1.8. Let h : EG −! f0; 1; : : : ; k − 1g, 2 ≤ k ≤ jEGj be a mapping that ∗ ∗ induces h : VG −! f0; 1; : : : ; k − 1g such that h (u) = h(e1)h(e2) : : : h(en)(mod k) for edges e1; e2; : : : ; en incident to u, then h is called k-total edge product cordial labeling if it satisfy js(a) − s(b)j ≤ 1 for a; b 2 f0; 1; : : : ; k − 1g. We refer to the articles [1, 3, 12] for more recent topics in labeling. In order to study different families of convex polytopes, first we give definitions of the following two archimedean convex polytopes introduced in [11]. Definitions 1.9. (1) A prism graph Ym is cartesian product graph Cm × P2, where Cm is cycle graph of order m and P2 is path graph of order 2 (see Figure 1). FIGURE 1. Prism graph Y4 (2) A m-sided anti-prism Am is polyhedron composed of two parallel copies of some particular m-sided polygon connected by alternating band of triangle (see Figure 2). FIGURE 2. Anti-prism graph A4 Baca introduced the following convex polytopes Rm in [6]. Definition 1.10. For m ≥ 5, a combination of prism graph Ym and antiprism graph Am is known as convex polytope graph Rm. It consists of the inner cycle vertices fui; 1 ≤ i ≤ mg, the middle cycle vertices fvi; 1 ≤ i ≤ mg and the outer cycle vertices fwi; 1 ≤ i ≤ mg 122 Umer Ali, Muhammad Bilal, Sohail Zafar and Zohaib Zahid (see Figure 3). Note that VRm = fui; vi; wi; 1 ≤ i ≤ mg and ERm = fuiui+1; 1 ≤ i ≤ m − 1g [ fvivi+1; 1 ≤ i ≤ m − 1g [ fwiwi+1; 1 ≤ i ≤ m − 1g [ fuivi; 1 ≤ i ≤ mg [ fui+1vi; 1 ≤ i ≤ m−1g[fviwi; 1 ≤ i ≤ mg[fumu1g[fvmv1g[fwmw1g[fu1vmg: w3 w2 v v3 2 u3 w4 w1 u4 u2 v1 v4 u5 u1 v8 v5 u6 u8 w5 w8 u7 v v6 7 w6 w7 FIGURE 3. Graph of convex polytope R8 Now we will discuss different families of convex polytopes obtained from Ym;Am and Rm. Definitions 1.11. (1) The graph of convex polytope Am can be obtained from Rm by adding some new lines (edges). i.e. VAm =VRm and EAm = ERm [ fvmw1g [ fviwi+1; 1 ≤ i ≤ m − 1g. Am consist of three-sided faces and a m-sides face (see Figure 4). w3 w4 v w2 v3 2 u3 u4 u2 v1 v4 w5 u5 u1 w1 v8 v5 u6 u8 u7 v v6 7 w6 w8 w7 FIGURE 4. Graph of convex polytope A8 (2) The graph of convex polytope Sm is composed of two parallel copies of prism graphs connected by alternating band of triangles. It consist of three-sided faces, four-sided faces and m-sides face (see Figure 5). Note that VG = fui; vi; wi; zi; 1 ≤ i ≤ mg and EG = fuiui+1; 1 ≤ i ≤ m − 1g [ fvivi+1; 1 ≤ i ≤ m − 1g [ fwiwi+1; 1 ≤ i ≤ m − 1g [ fzizi+1; 1 ≤ i ≤ m − 1g [ fuivi; 1 ≤ i ≤ mg [ fviwi; 1 ≤ i ≤ mg [ fvi+1wi; 1 ≤ i ≤ m − 1g [ fwizi; 1 ≤ i ≤ mg [ fumu1g [ fvmv1g [ fwmw1g [ fzmz1g [ fv1wmg: Some Families of Convex Polytopes Labeled by 3-Total Edge Product Cordial Labeling 123 z z3 2 w2 w3 v3 z v4 v2 1 z4 w1 u3 u2 w4 u4 u1 v5 v1 u5 u8 u w8 6 u w 7 5 v v z8 z5 6 8 v7 w7 w6 z z6 7 FIGURE 5. Graph of convex polytope S8 (3) The convex polytope Qm can be obtained from Sm by deleting some lines (edges). i.e. VQm =VSm and EQm = ESm n fwiwi+1; 1 ≤ i ≤ m − 1g [ fwmw1g. It consist of three-sided faces, four-sided faces, five-sided faces and m-sides face (see Figure 6). z3 z2 w2 w3 v3 z v4 v2 1 z4 w1 u3 u2 w4 u4 u1 v5 v1 u5 u 8 w u6 8 u7 w5 v v z8 z5 6 8 v7 w7 w6 z6 z7 FIGURE 6. Graph of convex polytope Q8 (4) The graph of convex polytope Tm can be obtained from Qm by adding some new lines (edges). i.e. VTm =VQm and ETm = EQm [ fui+1vi; 1 ≤ i ≤ mg [ fu1vmg. It consist of three-sided faces, five-sided faces and m-sides face (see Figure 7). 124 Umer Ali, Muhammad Bilal, Sohail Zafar and Zohaib Zahid z z3 2 w2 w3 v3 z v4 v2 1 z4 w1 u3 u2 w4 u4 u1 v5 v1 u5 u8 u w8 6 u w 7 5 v v z8 z5 6 8 v7 w7 w6 z z6 7 FIGURE 7.