Remainder Cordial Labeling of Graphs

Journal of Algorithms and Computation journal homepage: http://jac.ut.ac.ir Remainder Cordial Labeling of Graphs R. Ponraj 1, K. Annathurai and R. Kala 3 1 Department of Mathematics, Sri Paramakalyani College, Alwarkurichi 67 41, India. Department of Mathematics, Thiruvalluvar College,, Papanasam 67 45, India. 3 Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli 67 01, India. ABSTRACT In this paper we introduce remainder cordial labeling of graphs. Let G be a (p, q) graph. Let f : V (G) {1,,..., p} be a 1 1 map. For each edge uv assign the label r where r is the remainder when f(u) is divided by f(v) or f(v) is divided by f(u) according as f(u) f(v) or f(v) f(u). The functionf is called a remainder cordial labeling of G if e f (0) e f (1) 1 where e f (0) and e f (1) respectively denote the number of edges labeled with even integers and odd integers. A graph G with a remainder cordial labeling is called a remainder cordial graph. We investigate the remainder cordial behavior of path, cycle, star, bistar, crown, comb, wheel, complete bipartite K,n graph. Finally we propose a conjecture on complete graph K n. ARTICLE INFO Article history: Received 8, January 017 Received in revised form 18, May 017 Accepted 30 May 017 Available on line 01, June 017 Keyword: vertex equitable labeling, vertex Path, cycle, star, bistar, crown, comb, wheel, complete bipartite graph, complete graph graph. AMS subject Classification: 05C78. Corresponding author: R. Ponraj. Email: ponrajmaths@gmail.com kannathuraitvcmaths@gmail.com karthipyi91@yahoo.co.in Journal of Algorithms and Computation 49 issue 1, June 017, PP. 17-30

18 Ponraj / JAC 49 issue 1, June 017, PP. 17-30 1 Introduction In this paper we consider only finite and simple graphs. Let G 1, G respectively be (p 1, q 1 ), (p, q ) graphs. Let G 1 and G be two graphs with vertex sets V 1 and V and edge sets E 1 and E respectively. Then their join G 1 + G is the graph whose vertex set is V 1 V and edge set is E 1 E {uv : u V 1 and v V }. The graph W n = C n + K 1 is called a wheel. In a wheel, a vertex of degree 3 is called a rim vertex. A vertex which is adjacent to all the rim vertices is called the central vertex. The edges with one end incident with the rim and the other incident with the central vertex are called spokes. The corona of G 1 with G, G 1 G is the graph obtained by taking one copy of G 1 and p 1 copies of G and joining the i th vertex of G 1 with an edge to every vertex in the i th copy of G. Cahit [1] introduced the concept of cordial labeling of graphs. Ponraj et al. [4] introduced quotient cordial labeling of graphs. In [4, 5], Ponraj et al. investigate the quotient cordial labeling behavior of path, cycle, star, bistar, complete graph, subdivided star S(K 1,n ), subdivided bistar S(B n,n ) and union of some star related graphs. Motivated by this labeling, we introduce remainder cordial labeling of graphs. Also, in this paper we investigate the remainder cordial labeling behavior of path, cycle, star, bistar, crown, comb, wheel, complete bipartite graph etc. Finally we propose a conjecture for K n. Terms that are not defined here follows from Harary [3] and Gallian []. Remainder cordial labeling Definition.1. Let G be a (p, q) graph. Let f : V (G) {1,,..., p} be an injective map. For each edge uv assign the label r where r is the remainder when f(u) is divided by f(v) or f(v) is divided by f(u) according as f(u) f(v) or f(v) f(u). The function f is called a remainder cordial labeling of G if e f (0) e f (1) 1 where e f (0) and e f (1) respectively denote the number of edges labelled with even integers and odd integers. A graph G with a remainder cordial labeling is called a remainder cordial graph. Example.. A simple example of a remainder cordial graph is shown in Figure 1. Figure 1: First we investigate the remainder cordial labeling behavior of a path. Theorem.3. All paths are remainder cordial.

19 Ponraj / JAC 49 issue 1, June 017, PP. 17-30 Proof. Let P n be the path u 1 u... u n 1 u n. Case (i). n 0 (mod 4) Assign the labels 1,, 3,..., n consecutively to the vertices u 1, u,..., u n and then assign the labels n +, n + 4,..., n to the vertices u n +1, u n +,..., u n + n respectively. Finally 4 assign the labels n + 1, n + 3,..., n + n 1 to the remaining vertices consecutively. Case(ii). n 1 (mod 4) In this case assign the labels 1,, 3,..., n+1 consecutively to the vertices u 1, u,..., u n+1 respectively. Next consider the vertex u n+3 to the vertices u n+3 vertices u 3n+5, u 3n+9 4 4 Case(iii). n (mod 4), u n+5,..., u 3n+1 4,..., u n respectively.. Assign the label n+1 +, n+1 +,..., 3n+1 4,..., n 1 to the. Next assign the labels n+3, n+7 In this case assign the labels 1,, 3,..., n to the vertices u 1, u,..., u n labels n +, n + 4,..., n 1 to the vertices u n +1, u n +,..., u n labels n + 1, n + 3,..., n to the vertices u n + n 4 +1, u n + n Case(iv). n 3 (mod 4) Assign the integers 1,,..., n+1. Next assign the + n 4. Finally assign the,..., u n. 4 continuously to the vertices u 1, u,..., u n+1. Next assign the labels n+5, n+5 +,..., n 1 to the vertices u n+1 +1, u n+1 +,..., u n+1 + n 3. Finally, 4 assign the labels n+3, n+7,..., n to the remaining vertices u 3n 1 +1, u 3n 1 +,..., u n. 4 4 Next to be considered are cycle and the wheel. Theorem.4. All cycles C n are remainder cordial. Proof. Let C n be the cycle u 1 u u 3... u n u 1. Case(i). n is odd Clearly the vertex labeling of the path P n given in Theorem.3, is also a remainder cordial labeling of C n. Case(ii). n is even Assign the labels 1,,..., n, n + 1 to the vertices u 1, u,..., u n, u n +1. Subcase(i). n (mod 4) In this case assign the labels n + 3, n + 5,..., n to the vertices u n +, u n +3,..., u 3n Next assign the labels n +, n + 4,..., n 1 to the vertices u 3n +, u 3n +3,..., u n. 4 4 Subcase(ii). n 0 (mod 4) In this case assign the labels n +3, n +5,..., n 1 to the vertices u n +, u n +3,..., u 3n 4 Finally assign the labels n +, n + 4,..., n to the remaining vertices consecutively. Theorem.5. The wheel W n is remainder cordial for all even n. 4 +1. 4 +1. Proof. Let W n = C n + K 1, where C n is the cycle u 1 u u 3... u n u 1 and V (K 1 ) = u. We now assign the labels to the vertices of the wheel W n. Assign the label 1 to the central vertex. Next assign the labels, 3,..., n, n + 1 consecutively to the rim vertices u 1, u, u 3,..., u n. This vertex labeling f is clearly a remainder cordial labeling, since e f (0) = e f (1) = n. Illustration 1. An illustration of wheel W 8 for remainder cordial graph is shown in Figure.

0 Ponraj / JAC 49 issue 1, June 017, PP. 17-30 We now investigate the star. Figure : Theorem.6. The star K 1,n is remainder cordial for all values of n. Proof. Assign the label to central vertex and assign the labels 1, 3, 4,..., n, n + 1 to the pendant vertices. Clearly this vertex labeling is a remainder cordial labeling. Next our focus of attention is the bistar. Theorem.7. The bistar B m,n is remainder cordial. Proof. Let V (B m,n ) = {u, v, u i, v j : 1 i m, 1 j n} and E(B m,n ) = {uv, uu i, vv j : 1 i m, 1 j n}. Clearly B m,n has m + n vertices and m + n + 1 edges. Assign the labels, 4 respectively to the vertices u, v. Next we move to the vertices u i. Assign the labels 1, 3, 5, 6,..., m + respectively to the vertices u 1, u,..., u m+. Finally assign the labels m + 3, m + 4,..., m + n to the remaining vertices. Table 1, establishes that this labeling f is a remainder cordial labeling. Nature of m and n e f (0) e f (1) m+n m+n m and n are of same parity + 1 m and n are of different parity Table 1: m+n+1 m+n+1 Next we investigate the crowns and the comb. Theorem.8. All crowns C n K 1 are remainder cordial. Proof. Let C n be the cycle u 1 u u 3... u n u 1 and v i (1 i n) be the pendant vertex adjacent to u i. Assign the labels 1, 3, 5,..., n 1 respectively to the vertices u 1, u 3, u 5,..., u n 1. Next move to the pendant vertices. Assign the even labels, 4,..., n to the vertices v 1, v,..., v n respectively. Clearly this vertex labeling f is a remainder cordial labeling, since e f (0) = e f (1) = n.

1 Ponraj / JAC 49 issue 1, June 017, PP. 17-30 Corollary.9. The Comb P n K 1 is remainder cordial. Proof. The labeling given in Theorem.8 obviously a remainder cordial labeling of P n K 1. Now we investigate the complete bipartite graph K,n. Theorem.10. The complete bipartite graph K,n is remainder cordial for all even values of n. Proof. Let V (K,n ) = {u, v, u i : 1 i n} and E(K,n ) = {uu i, vv i : 1 i n}. Assign the labels, 4 respectively to the vertices u, v and assign the labels 1, 3, 5, 6,..., n to the remaining non-labelled vertices in any order. Clearly this vertex labeling f is a remainder cordial labeling, since e f (0) = e f (1) = n. Our next investigation is about the complete graph K n. According to the visual basic program.11 and its output.1 it is seen that, K n (4 n 510) is not remainder cordial. Program.11. SORT OUT REMAINDER ODD OR EVEN CODING Dim con As New ADODB.Connection Dim rs As New ADODB.Recordset Private Sub Command1_Click() If rs.bof And rs.eof Then rs.addnew rs.movelast Dim no,c1, f, c, gno, gno, n, odd, eve, div, i, j, k, s As Double c1 = 0 c = 0 f = 0 odd = 0 eve = 0 gno = Val(Text1.Text) gno = Val(Text.Text) n = gno For no = gno To gno Step 1 hide these 6 statement if u dont want listbox List1.AddItem "-----------------------" List1.AddItem "---For Number = " & n

Ponraj / JAC 49 issue 1, June 017, PP. 17-30 List1.AddItem "-----------------------" List.AddItem "-----------------------" List.AddItem "---For Number = " & n List.AddItem "-----------------------" For i = n To Step -1 k = i - 1 For j = k To 1 Step -1 s = i Mod j If s Mod = 0 Then eve = eve + 1 hide these 3 statement if u dont want listbox List1.AddItem i & " % " & j & " = " & s List1.Visible = True lbleven.visible = True c1 = c1 + 1 Else odd = odd + 1 hide these 3 statement if u dont want listbox List.AddItem i & " % " & j & " = " & s List.Visible = True lblodd.visible = True c = c + 1 End If Next j Next i c3 = c1 - c rs.movefirst Do While Not rs.eof If n = rs(0) Then f = 1 Exit Do End If rs.movenext Loop If found = 0 Then rs.movelast rs.addnew rs(0) = n rs(1) = c1 rs() = c

3 Ponraj / JAC 49 issue 1, June 017, PP. 17-30 rs(3) = c3 rs.update End If n = n - 1 c1 = 0 c = 0 c3 = 0 f = 0 Next no MsgBox "recorded" End Sub Private Sub Command_Click() List1.Clear List.Clear Text1.Text = "" List1.Visible = False List.Visible = False lblodd.visible = False lbleven.visible = False Text1.SetFocus End Sub Private Sub Command3_Click() End End Sub Private Sub Form_Load() con.open "Provider=Microsoft.Jet.OLEDB.4.0; Data Source=D:\genoddeven\result.mdb;Persist Security Info=False;" rs.open "select * from res", con, adopendynamic, adlockoptimistic End Sub Output.1. n e f (0) e f (1) e f (0) e f (1) n e f (0) e f (1) e f (0) e f (1) 1 0 0 0 1 0 1

4 Ponraj / JAC 49 issue 1, June 017, PP. 17-30 3 1 1 4 4 5 6 4 6 10 5 5 7 1 9 3 8 17 11 6 9 1 15 6 10 7 18 9 11 31 4 7 1 39 7 1 13 43 35 8 14 53 38 15 15 59 46 13 16 68 5 16 17 75 61 14 18 87 66 1 19 93 78 15 0 107 83 4 1 115 95 0 19 10 7 3 137 116 1 4 153 13 30 5 16 138 4 6 179 146 33 7 190 161 9 8 07 171 36 9 18 188 30 30 39 196 43 31 48 17 31 3 70 6 44 33 8 46 36 34 304 57 47 35 318 77 41 36 341 89 5 37 354 31 4 38 379 34 55 39 394 347 47 40 419 361 58 41 434 386 48 4 463 398 65 43 476 47 49 44 507 439 68 45 54 466 58 46 553 48 71 47 570 511 59 48 601 57 74 49 619 557 6 50 65 573 79 51 671 604 67 5 704 6 8 53 73 655 68 54 760 671 89 55 779 706 73 56 816 74 9 57 837 759 78 58 874 779 95 59 895 816 79 60 936 834 10 61 955 875 80 6 998 893 105 63 101 93 89 64 1061 955 106 65 1087 993 94 66 119 1016 113 67 1153 1058 95 68 1197 1081 116 69 13 113 100 70 169 1146 13 71 193 119 101 7 134 114 18 73 1365 163 10 74 1416 185 131 75 1443 133 111 76 149 1358 134 77 151 1405 116 78 157 1431 141 79 1599 148 117 80 165 1508 144 81 168 1558 14 8 1734 1587 147 83 1764 1639 15 84 180 1666 154

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6 Ponraj / JAC 49 issue 1, June 017, PP. 17-30 167 7085 6776 309 168 7195 6833 36 169 754 694 31 170 7367 6998 369 171 748 7107 31 17 7539 7167 37 173 7600 778 3 174 7715 7336 379 175 7778 7447 331 176 7891 7509 38 177 7956 760 336 178 8069 7684 385 179 8134 7797 337 180 853 7857 396 181 8314 7976 338 18 8437 8034 403 183 8498 8155 343 184 861 815 406 185 8684 8336 348 186 8809 8396 413 187 887 8519 353 188 8997 8581 416 189 9066 8700 366 190 9189 8766 43 191 956 8889 367 19 9381 8955 46 193 9448 9080 368 194 9575 9146 49 195 9648 967 381 196 977 9338 434 197 9844 946 38 198 9974 959 445 199 1004 9659 383 00 10175 975 450 01 1044 9856 388 0 10377 994 453 03 10448 10055 393 04 10583 1013 460 05 10654 1056 398 06 10789 1036 463 07 10864 10457 407 08 10997 10531 466 09 11074 1066 41 10 1113 1073 481 11 1184 10871 413 1 1145 10941 484 13 11498 11080 418 14 11639 1115 487 15 11714 1191 43 16 11857 11363 494 17 1193 11504 48 18 1075 11578 497 19 115 11719 433 0 197 11793 504 1 1374 11936 438 151 1010 511 3 1596 1157 439 4 1745 131 514 5 187 1373 454 6 1971 1454 517 7 13053 1598 455 8 1301 1677 54 9 1381 185 456 30 13433 190 531 31 13517 13048 469 3 13665 13131 534 33 13749 1379 470 34 13903 13358 545 35 13985 13510 475 36 14139 13591 548 37 143 13743 480 38 14379 1384 555 39 14461 13980 481 40 1461 14059 56 41 14701 1419 48 4 14864 1497 567 43 14947 14456 491 44 15108 14538 570 45 15195 14695 500 46 15356 14779 577 47 15443 14938 505 48 15604 1504 580

7 Ponraj / JAC 49 issue 1, June 017, PP. 17-30 49 15693 15183 510 50 15856 1569 587 51 15943 1543 511 5 1611 15514 598 53 16197 15681 516 54 16366 15765 601 55 16457 1598 59 56 1661 16019 60 57 16713 16183 530 58 16881 167 609 59 16973 16438 535 60 17143 1657 616 61 1737 16693 544 6 17405 16786 619 63 17499 16954 545 64 17671 17045 66 65 17765 1715 550 66 17939 17306 633 67 18033 17478 555 68 1807 17571 636 69 18301 17745 556 70 18483 1783 651 71 18571 18014 557 7 18755 18101 654 73 18849 1879 570 74 1909 1837 657 75 1917 18548 579 76 19307 18643 664 77 19403 1883 580 78 19585 18918 667 79 19685 19096 589 80 19867 19193 674 81 19965 19375 590 8 0151 19470 681 83 047 19656 591 84 0435 19751 684 85 0537 19933 604 86 073 003 691 87 085 016 609 88 101 0316 696 89 1114 050 61 90 1304 0601 703 91 1406 0789 617 9 1596 0890 706 93 1698 1080 618 94 1894 1177 717 95 1994 1371 63 96 190 1470 70 97 96 1660 636 98 488 1765 73 99 596 1955 641 300 79 058 734 301 898 5 646 30 3094 357 737 303 30 551 651 304 3398 658 740 305 3508 85 656 306 3708 957 751 307 3814 3157 657 308 4018 360 758 309 414 346 66 310 4330 3565 765 311 4434 3771 663 31 4644 387 77 313 4746 408 664 314 4958 4183 775 315 5070 4385 685 316 574 4496 778 317 5386 4700 686 318 5594 4809 785 319 5706 5015 691 30 5914 516 788 31 608 533 696 3 638 5443 795 33 635 5651 701 34 6565 5761 804 35 6680 5970 710 36 6891 6084 807 37 7008 693 715 38 719 6409 810 39 7338 6618 70 330 7555 6730 85

8 Ponraj / JAC 49 issue 1, June 017, PP. 17-30 331 7668 6947 71 33 7887 7059 88 333 8004 774 730 334 81 7390 831 335 8340 7605 735 336 8559 771 838 337 8676 7940 736 338 8898 8055 843 339 9016 875 741 340 940 8390 850 341 9358 861 746 34 9586 875 861 343 970 8951 751 344 9930 9066 864 345 3005 988 764 346 3076 9409 867 347 30398 9633 765 348 3066 975 874 349 30746 9980 766 350 30980 30095 885 351 3110 3033 779 35 3133 30444 888 353 31454 30674 780 354 31688 30793 895 355 31810 3105 785 356 3044 31146 898 357 317 31374 798 358 340 31501 901 359 3530 31731 799 360 3766 31854 91 361 3891 3089 80 36 3318 313 915 363 3357 3446 811 364 33494 357 9 365 3363 3807 816 366 3386 3933 99 367 33989 3317 817 368 3430 3398 93 369 34361 33535 86 370 3460 33663 939 371 34733 3390 831 37 34976 34030 946 373 35105 3473 83 374 3535 34399 953 375 35485 34640 845 376 3578 3477 956 377 35863 35013 850 378 3611 35141 971 379 3641 35390 851 380 36494 35516 978 381 3663 35767 856 38 36876 35895 981 383 37005 36148 857 384 3760 3676 984 385 37395 3655 870 386 37646 36659 987 387 37785 36906 879 388 38034 37044 990 389 38173 3793 880 390 38430 3745 1005 391 38565 37680 885 39 3883 37813 1010 393 38959 38069 890 394 3917 3804 1013 395 39355 38460 895 396 39617 38593 104 397 39751 38855 896 398 40015 38988 107 399 40155 3946 909 400 40416 39384 103 401 40555 39645 910 40 4080 39781 1039 403 40959 40044 915 404 414 4018 104 405 41371 40439 93 406 4163 40583 1049 407 41779 4084 937 408 404 40986 1056 409 4187 4149 938 410 4454 41391 1063 411 4599 41656 943 41 4866 41800 1066

9 Ponraj / JAC 49 issue 1, June 017, PP. 17-30 413 43013 4065 948 414 4384 407 1077 415 4349 4476 953 416 43700 460 1080 417 43847 4889 958 418 4410 43033 1087 419 4465 43306 959 40 44546 43444 110 41 44685 4375 960 4 44968 43863 1105 43 45111 4414 969 44 4539 4484 1108 45 45539 44561 978 46 4580 44705 1115 47 45967 44984 983 48 4648 45130 1118 49 46401 45405 996 430 46680 45555 115 431 46831 45834 997 43 47114 4598 113 433 4763 4665 998 434 47550 46411 1139 435 47703 4669 1011 436 47986 46844 114 437 48141 4715 1016 438 4846 4777 1149 439 48579 4756 1017 440 48868 4771 1156 441 4906 47994 103 44 4931 48149 1163 443 49468 48435 1033 444 49758 48588 1170 445 49914 48876 1038 446 5004 49031 1173 447 5036 49319 1043 448 5065 49476 1176 449 50810 49766 1044 450 51109 49916 1193 451 516 5013 1049 45 51561 50365 1196 453 51716 5066 1054 454 5015 50816 1199 455 5176 51109 1067 456 5473 5167 106 457 563 51564 1068 458 5931 517 109 459 53096 5015 1081 460 53393 5177 116 461 53556 5474 108 46 53861 5630 131 463 54018 5935 1083 464 5435 53091 134 465 54488 5339 1096 466 54791 53554 137 467 54954 53857 1097 468 5563 54015 148 469 5544 543 110 470 55735 54480 155 471 55896 54789 1107 47 5607 54949 158 473 56370 5558 111 474 56683 55418 165 475 56848 5577 111 476 57161 55889 17 477 5738 56198 1130 478 57639 56364 175 479 57806 56675 1131 480 5811 56839 18 481 5888 5715 1136 48 58603 57318 185 483 58776 5767 1149 484 59088 57798 190 485 596 58108 1154 486 59578 5877 1301 487 59748 58593 1155 488 60066 5876 1304 489 6038 59078 1160 490 60560 5945 1315 491 6078 59567 1161 49 61054 5973 13 493 61 60056 1166 494 61550 601 139

30 Ponraj / JAC 49 issue 1, June 017, PP. 17-30 495 6176 60539 1187 496 6046 60714 133 497 64 6103 119 498 6546 6107 1339 499 67 6159 1193 500 63048 6170 1346 501 634 606 1198 50 63550 601 1349 503 6376 657 1199 504 64058 6698 1360 505 643 6308 104 506 64566 63199 1367 507 6474 6359 113 508 65074 63704 1370 509 6550 64036 114 510 65590 6405 1385 The output.1 naturally leads us to propose the following conjecture. Conjecture.13. The complete graph K n is remainder cordial iff n 3. Acknowledgement. We thank to the referees for their valuable suggestions. References [1] Cahit, I., Cordial Graphs : A weaker version of Graceful and Harmonious graphs, Ars combin., 3 (1987), 01 07. [] Gallian, J.A., A Dynamic survey of graph labeling, The Electronic Journal of Combinatorics., 18, (016). [3] Harary, F., Graph theory, Addision Wesley, New Delhi, 1969. [4] Ponraj, R., Maria Adaickalam, M., and Kala, R., Quotient cordial labeling of graphs, International Journal of Math. Combin., Vol.1, (016), 101 108. [5] Ponraj, R. and Maria Adaickalam, M., Quotient cordial labeling of some star related graphs, The Journal of the Indian Academy of Mathematics, 37(), (015), 313 34.