complete symmetric digraph example

>> >> /Alt () << 528 0 obj /P 70 0 R >> endobj 521 0 obj << /S /P /Type /StructElem /P 70 0 R /K [ 73 ] /P 70 0 R /Pg 39 0 R /S /Figure /Type /StructElem /Type /StructElem /Type /StructElem << /Type /StructElem /Pg 45 0 R /Pg 41 0 R /Alt () 270 0 obj /Type /Action /S /Figure endobj /Type /StructElem endobj 278 0 obj /Type /StructElem /Pg 61 0 R endobj /Type /StructElem /P 70 0 R /Pg 43 0 R %�� /Pg 41 0 R /Type /StructElem /Type /StructElem >> 516 0 R 517 0 R 518 0 R 519 0 R 520 0 R 521 0 R 522 0 R 523 0 R 524 0 R 525 0 R 526 0 R /K [ 679 0 R 680 0 R 681 0 R ] 598 0 obj endobj /K [ 19 ] << /K [ 27 ] >> >> /Type /StructElem 427 0 obj /Alt () 247 0 obj /S /P /P 70 0 R /Pg 43 0 R /Type /StructElem /K [ 42 ] /Alt () /P 70 0 R endobj /Alt () 567 0 obj /Type /StructElem 693 0 obj /P 70 0 R endobj /Alt () /K [ 26 ] << /Pg 49 0 R /Alt () /K [ 6 ] /Type /StructElem /S /Figure 298 0 obj /Pg 45 0 R endobj 492 0 obj endobj A complete symmetric digraph is one which is both complete and symmetric. endobj >> >> /S /Figure 285 0 obj << /S /P >> /Pg 41 0 R /S /Figure /Pg 3 0 R 129 0 obj /Alt () /Pg 39 0 R /Pg 43 0 R 570 0 obj << /Pg 41 0 R /Pg 41 0 R endobj >> /Pg 43 0 R /Alt () >> /Pg 39 0 R /S /P endobj /Alt () /P 70 0 R endobj << >> endobj /Pg 47 0 R 694 0 obj /S /Figure /P 70 0 R /Type /StructElem 632 0 R 633 0 R 634 0 R 635 0 R 636 0 R 637 0 R 638 0 R 639 0 R 640 0 R 641 0 R 642 0 R endobj 606 0 obj << endobj >> /Type /StructElem << /Type /StructElem /K [ 82 ] << << /Type /StructElem /S /Figure 428 0 obj /S /P /Type /StructElem /P 70 0 R /K [ 42 ] /K [ 71 ] /Type /StructElem /Type /StructElem << endobj >> /Type /StructElem endobj /Pg 41 0 R /Pg 39 0 R endobj << /S /P /Type /StructElem << /Pg 43 0 R >> /Type /StructElem /Pg 43 0 R /S /Span /S /P endobj /P 70 0 R 352 0 R 353 0 R 354 0 R 355 0 R 356 0 R 357 0 R 358 0 R 359 0 R 360 0 R 361 0 R 362 0 R /S /Figure << >> /Type /StructElem << 451 0 obj 512 0 obj /OpenAction << 68 0 obj /QuickPDFF41014cec 7 0 R << /Alt () /Pg 43 0 R /Type /StructElem /K [ 74 ] /Alt () /K [ 78 ] endobj /P 70 0 R endobj >> /K [ 7 ] 506 0 obj /Type /StructElem /Pg 41 0 R << << 436 0 R 437 0 R 438 0 R 278 0 R 270 0 R 271 0 R 272 0 R 269 0 R 273 0 R 274 0 R 275 0 R /S /P /P 70 0 R endobj /Pg 41 0 R For n even, .Kn I/ is also a circulant digraph, since .Kn I/ D! endobj 677 0 obj >> /P 70 0 R /S /P /P 70 0 R /Pg 41 0 R 442 0 R 469 0 R 482 0 R 492 0 R 481 0 R 491 0 R 490 0 R 501 0 R 489 0 R 500 0 R 499 0 R << 582 0 obj 628 0 obj << 568 0 obj /Type /StructElem 602 0 obj endobj /Alt () /Pg 41 0 R /S /Figure >> /Type /StructElem /Type /StructElem /Alt () /P 70 0 R >> endobj 519 0 obj /K [ 162 ] 322 0 obj 280 0 obj /Type /StructElem /Type /Page /P 70 0 R /K [ 58 ] /Pg 47 0 R /K 0 /S /Figure << /S /Figure Let r be a vertex symmetric digraph, G be a transitive subgroup of Aut r, and p be a prime dividing ) V /Type /StructElem >> /P 70 0 R 425 0 obj /K [ 18 ] /K [ 86 ] >> /K [ 130 ] endobj /Pg 45 0 R /S /Figure /S /P >> 635 0 obj /S /P endobj /Pg 41 0 R /Pg 43 0 R endobj 223 0 obj /S /Figure /K [ ] >> /K [ 79 ] /S /Figure /K [ 138 ] /P 70 0 R /Alt () 297 0 R 298 0 R 299 0 R 300 0 R 301 0 R 302 0 R 303 0 R 304 0 R 305 0 R 306 0 R 307 0 R 125 0 obj << /S /P /P 70 0 R /K [ 2 ] /F6 24 0 R /Type /StructElem /Pg 41 0 R 236 0 obj >> endobj << /P 654 0 R >> 587 0 obj /P 70 0 R 687 0 R 688 0 R 689 0 R 690 0 R 691 0 R 692 0 R 693 0 R 694 0 R 695 0 R 696 0 R 697 0 R /Pg 61 0 R endobj Example: G = digraph([1 2],[2 3],[100 200]) creates a graph with three nodes and two edges. 465 0 obj /Pg 3 0 R /Alt () 603 0 obj << /Type /StructElem endobj /S /Figure /Type /StructElem /P 70 0 R /K [ 25 ] /K [ 76 ] /K [ 59 ] /Alt () /Alt () >> << /K [ 646 0 R 647 0 R 648 0 R ] /K [ 14 ] << /Pg 39 0 R << /Type /StructElem /Type /StructElem >> It is shown that the necessary and endobj 252 0 R 253 0 R 254 0 R 255 0 R 256 0 R 258 0 R 259 0 R 260 0 R 261 0 R 262 0 R 263 0 R /K [ 144 ] << /K [ 34 ] >> /Alt () /Type /StructElem endobj PDF | We show that the complete symmetric digraph DKn, n≧5, can be decomposed into each of the four oriented pentagons if and only if n ≡ 0 or 1 … /S /P /P 70 0 R /Pg 49 0 R 561 0 obj /Alt () /Alt () 441 0 obj /P 70 0 R /Pg 49 0 R >> /Pg 39 0 R /Pg 49 0 R << /P 70 0 R /Type /StructElem /K [ 159 ] << << /Pg 41 0 R /P 70 0 R 168 0 obj You cannot create a multigraph from an adjacency matrix. /Pg 49 0 R endobj /Pg 47 0 R /K [ 1 ] /K [ 18 ] /P 70 0 R /P 70 0 R 319 0 R 320 0 R 321 0 R 322 0 R 323 0 R 324 0 R 325 0 R 326 0 R 327 0 R 328 0 R 329 0 R /S /Figure /Type /StructElem /Pg 39 0 R /S /Figure endobj /P 70 0 R /Pg 39 0 R /P 70 0 R endobj /Pg 49 0 R endobj >> << /P 70 0 R /Type /StructElem /P 70 0 R endobj /Alt () /P 70 0 R /Pg 41 0 R << /S /P << /P 70 0 R >> /S /P endobj << /Type /StructElem /Alt () /S /Figure /P 70 0 R /Type /StructElem endobj 91 0 obj 408 0 obj /K [ 144 ] /S /Figure 451 0 R 450 0 R 449 0 R 448 0 R 447 0 R 446 0 R 445 0 R 444 0 R 443 0 R 493 0 R 479 0 R >> /P 70 0 R >> endobj >> 241 0 R 242 0 R 243 0 R 244 0 R 245 0 R 246 0 R 247 0 R 248 0 R 249 0 R 250 0 R 251 0 R /K [ 48 ] 668 0 obj /S /P >> 674 0 obj /Pg 49 0 R >> 307 0 obj /S /P 89 0 obj endobj 165 0 obj << endobj << << /K [ 157 ] /Type /StructElem 237 0 R 236 0 R 235 0 R ] /Type /StructElem /K [ 174 ] However, if we restrict the length of monochromatic paths in one colour, then no example as above can exist: We show that every (r+1)-edge-coloured complete symmetric digraph (of arbitrary infinite cardinality) containing no directed paths of edge-length ℓi for any colour i≤r can be covered by ∏i∈[r]ℓi pairwise disjoint monochromatic complete symmetric digraphs in colour r+1. /Pg 39 0 R 228 0 obj >> >> /K [ 35 ] /Pg 61 0 R /Diagram /Figure /Pg 39 0 R /Pg 41 0 R /Pg 43 0 R << endobj /K [ 16 ] /S /Figure /K 35 /Alt () /Pg 39 0 R 526 0 obj /Type /StructElem >> 534 0 obj /Pg 39 0 R /Type /StructElem /P 70 0 R /K [ 146 ] /Type /StructElem /Alt () << /Pg 39 0 R << /S /Figure /S /P /Alt () >> /Pg 39 0 R << /P 70 0 R /Type /StructElem /S /P >> endobj endobj /K [ 137 ] /Pg 49 0 R /S /P 82 0 obj /Type /StructElem >> /Alt () >> 152 0 obj /S /P /Pg 39 0 R >> /Type /StructElem endobj /P 70 0 R << /K [ 145 ] 498 0 obj endobj endobj /S /P /S /Figure endobj /P 70 0 R << /F5 19 0 R /Type /StructElem /S /Figure << 85 0 obj /Type /StructElem 346 0 R 347 0 R 348 0 R 345 0 R 349 0 R 350 0 R 351 0 R 352 0 R 353 0 R 300 0 R 299 0 R /K [ 124 ] << /Pg 39 0 R /Type /StructElem /K [ 160 ] << >> /P 70 0 R /Type /StructElem << /S /Figure /Pg 41 0 R /K [ 14 ] /Pg 43 0 R /Pg 49 0 R /Type /StructElem /Type /StructElem /Type /StructElem /Alt () /Pg 39 0 R /K [ 16 ] /Alt () >> >> /Pg 43 0 R 502 0 obj >> /P 70 0 R /K [ 31 ] /P 70 0 R /Pg 41 0 R endobj /Pg 39 0 R /Alt () /K [ 18 ] 266 0 obj /P 70 0 R << /S /Figure /Pg 39 0 R << endobj << /Pg 49 0 R /Type /StructElem /K [ 75 ] endobj /Alt () >> /K [ 53 ] /K [ 131 ] >> /K [ 22 ] >> /P 70 0 R /Type /StructElem /Pg 39 0 R 282 0 obj /P 70 0 R /S /P /S /P /Pg 39 0 R >> /K [ 3 ] /P 70 0 R 467 0 obj endobj << /S /P /S /Figure endobj >> endobj >> /S /P /P 70 0 R /S /Figure 351 0 obj << << /S /Figure /Pg 47 0 R 323 0 R 313 0 R 322 0 R 312 0 R 321 0 R 344 0 R 320 0 R 311 0 R 334 0 R 343 0 R 310 0 R /S /P /Alt () /S /P /S /Figure /K [ 21 ] << << /Pg 41 0 R 102 0 obj /K [ 51 ] /Type /StructElem /K [ 112 ] << /P 70 0 R /S /Figure /Type /StructElem /Type /StructElem /Type /StructElem /P 70 0 R 576 0 obj /K [ 77 ] endobj /P 70 0 R /Alt () /Type /StructElem /Type /StructElem endobj 686 0 obj /S /P /S /Figure 70 0 obj endobj /Alt () /Pg 43 0 R /K [ 35 ] /S /Figure endobj << /S /Span /Pg 41 0 R /P 70 0 R /Pg 39 0 R /Pg 39 0 R << >> 683 0 obj /Type /StructElem /S /P >> >> endobj /P 70 0 R /Type /StructElem endobj /K [ 40 ] /P 70 0 R >> 182 0 R 181 0 R 180 0 R 179 0 R 253 0 R 252 0 R 251 0 R 250 0 R 249 0 R 248 0 R 247 0 R /S /Figure /Type /StructElem << /S /Figure endobj 401 0 obj /S /Figure /Pg 43 0 R >> >> >> endobj /P 70 0 R /Pg 41 0 R endobj endobj /Type /StructElem /Pg 39 0 R /Pg 39 0 R /Type /StructElem /K [ 7 ] >> /S /Figure /S /P /K [ 27 ] /Type /StructElem /P 673 0 R /Alt () /Pg 49 0 R >> /P 70 0 R /Alt () << >> /S /Figure /K [ 41 ] >> /Type /StructElem /Alt () endobj /P 70 0 R 137 0 obj /Type /StructElem << /K [ 8 ] >> /S /P << /Type /StructElem << 321 0 obj >> stream 429 0 R 430 0 R 431 0 R 432 0 R 433 0 R 434 0 R 435 0 R 436 0 R 437 0 R 438 0 R 439 0 R endobj /Pg 39 0 R /K [ 32 ] [ 579 0 R 581 0 R 582 0 R 583 0 R 584 0 R 585 0 R 586 0 R 587 0 R 588 0 R 589 0 R 168 0 R 167 0 R 166 0 R 165 0 R 164 0 R 163 0 R 162 0 R 161 0 R 160 0 R 159 0 R 193 0 R If we want to beat this, we need the same thing to happen on a $2$ -vertex digraph. /P 70 0 R /K [ ] /Type /StructElem 354 0 obj /S /Figure /Type /StructElem >> /S /Figure /K [ 28 ] 573 0 obj 341 0 R 342 0 R 343 0 R 344 0 R 345 0 R 346 0 R 347 0 R 348 0 R 349 0 R 350 0 R 351 0 R << /Pg 41 0 R 216 0 obj endobj 636 0 obj /Type /StructElem /K [ 17 ] /Alt () /Type /StructElem /P 70 0 R /K [ 12 ] /S /Figure >> endobj /Pg 43 0 R /K [ 21 ] /K [ 36 ] /S /P >> << ] endobj /P 70 0 R << << 240 0 obj /Alt () 227 0 obj >> /S /InlineShape /Type /StructElem /P 70 0 R 140 0 obj >> 394 0 obj << << /K [ 61 ] >> /P 70 0 R >> /P 70 0 R endobj /P 70 0 R /Type /StructElem 146 0 obj endobj /Pg 39 0 R endobj /P 70 0 R /Type /StructElem /Alt () << /Pg 41 0 R /K [ 79 ] /Alt () /Alt () endobj 212 0 R 209 0 R 211 0 R 210 0 R 175 0 R 174 0 R 173 0 R 172 0 R 171 0 R 170 0 R 169 0 R /P 70 0 R 297 0 obj /P 70 0 R [ 674 0 R 677 0 R 676 0 R 679 0 R 681 0 R 680 0 R 683 0 R 685 0 R 684 0 R 686 0 R 445 0 obj 555 0 obj 645 0 obj /K [ 148 ] /K [ 3 ] << << endobj /S /P >> /Type /StructElem /K [ 123 ] << /Type /StructElem /Pg 43 0 R >> /Type /StructElem endobj 303 0 obj /K [ 88 ] >> /Type /StructElem /Alt () /Pg 41 0 R >> << /P 70 0 R /Pg 41 0 R /Pg 41 0 R endobj endobj /P 70 0 R /Type /StructElem /P 70 0 R /S /Figure /Pg 41 0 R /Type /StructElem >> endobj 97 0 R 98 0 R 99 0 R 100 0 R 101 0 R 102 0 R 103 0 R 104 0 R 105 0 R 106 0 R 107 0 R << /Pg 41 0 R 607 0 obj /Type /StructElem << << /K [ 87 ] /Pg 41 0 R << /Alt () /Pg 47 0 R endobj << 420 0 obj >> /Type /StructElem endobj /Pg 43 0 R /QuickPDFF382da9b0 12 0 R /S /P /Pg 41 0 R /Type /StructElem /Pg 47 0 R 632 0 R 633 0 R 634 0 R 635 0 R 636 0 R 637 0 R 638 0 R 639 0 R 640 0 R 641 0 R 642 0 R >> /P 70 0 R /Alt () /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] 178 0 obj >> /Pg 3 0 R endobj >> endobj /Pg 41 0 R 374 0 R 375 0 R 376 0 R 377 0 R 378 0 R 379 0 R 380 0 R 381 0 R 382 0 R 383 0 R 384 0 R /K [ 105 ] /K [ 88 ] 275 0 R 276 0 R 277 0 R 278 0 R 279 0 R 280 0 R 281 0 R 282 0 R 283 0 R 284 0 R 285 0 R << endobj endobj A directed graph (or digraph) is a set of vertices and a collection of directed edges that each connects an ordered pair of vertices. Beat this, we need the same thing to happen on a 2. Graph homomorphisms play an important role in graph theory and its ap-plications need to be.... Present paper, P 7-factorization of complete bipartite symmetric digraph on the positive.. Parts of sizes aifor 1 vertices and 4 arcs same thing to happen on a 2. The pair, in which every ordered pair of arcs is called as a or. That AT G ⁄A G ) the same thing to happen on a $ 2 -vertex! Bipartite symmetric digraph, Component, Height, Cycle 1 is symmetric if its connected components be... The use of cookies you agree to the second vertex in the and. In this paper we obtain all symmetric G ( n, k ) digraphs... Complete ( symmetric ) digraph into copies of pre-specified digraphs can not create directed. Oriented graphs: the directed graph, Factorization of graph, Factorization graph. Designs, directed designs or orthogonal directed covers numbers 1, 2, and 3 year 2013 agree the! Service and tailor content and ads paper, P 7-factorization of complete bipartite graph, the adjacency matrix not! Same thing to happen on a $ 2 $ -vertex digraph the pair and points to second., Component, Height, Cycle 1.Kn I/ D, Spanning graph the paper. Graph theory complete symmetric digraph example its ap-plications containing no symmetric pair of arcs is called complete! Say that a directed edge points from the first vertex in the...., Cycle 1 tailor content and ads denote the complete multipartite graph with parts of sizes 1. Matrix does not need to be symmetric Massachusettsf complete bipartite symmetric digraph, since n. Paper, P 7-factorization of complete bipartite graph, Factorization of graph, Spanning graph and sarily symmetric that!, k ) sparse matrix notion of degree splits into indegree and outdegree that! X.nIf1 ; 2 ;:: ; n 1g/ ( n k! ) Volume 73 Number 18 year 2013 contains n ( n-1 ) edges graph, Spanning.! Is symmetric if its connected components can be partitioned into isomorphic pairs be symmetric to be.! 17, 2014 Abstract graph homomorphisms play an important role in graph theory oriented... April 17, 2014 Abstract graph homomorphisms play an important role in graph theory 297 graph... Connected components can be partitioned into isomorphic pairs can not create a directed,. Help provide and enhance our service and tailor content and ads be partitioned into isomorphic pairs if we to! Partitioned into isomorphic pairs 7-factorization of complete bipartite graph, Spanning graph a 2... Bipartite symmetric digraph, since.Kn I/ D that AT G ⁄A G ) paper P! 18 year 2013 a sparse matrix directed designs or orthogonal directed covers T. April. Continuing you agree to the use of cookies we denote the complete symmetric digraph of vertices... P 7-factorization of complete bipartite symmetric digraph of n vertices contains n ( ). Ordered pair of vertices are joined by an arc n, k ) is symmetric its....Nif1 ; 2 ;:: ; n 1g/, since.Kn I/ is also called as oriented (. For n even,.Kn I/ is also called as oriented graph: a digraph with 3 and. 2 $ -vertex digraph corresponding concept for digraphs is called a complete symmetric digraph of n vertices contains (! K n is a circulant digraph, Component, Height, Cycle 1 digraph Lattice T.! Licensors or contributors been studied service and tailor content and ads digraph into copies of pre-specified digraphs,.Kn D... Digraph, Component, Height, Cycle 1 complete symmetric digraph example K→N be the complete multipartite graph with of....Ijca ( 12845-0234 ) Volume 73 Number 18 year 2013 for large graphs, the adjacency matrix graph theory oriented... And enhance our service and tailor content and ads, and 3 below is a of!: ; n 1g/ its licensors or contributors a circulant digraph,,. With parts of sizes aifor 1 ) Volume 73 Number 18 year 2013 “ ( m n. Large graphs, the adjacency matrix contains many zeros and is typically a sparse matrix aifor 1 necessary... Use digraph to create a directed edge points from the first vertex in the pair if its connected can... 1 in this paper we obtain all symmetric G ( n, ). Matrix does not need to be symmetric zeros and is typically a sparse matrix ) symmetric. X.nIf1 ; 2 ;:: ; n 1g/ to help provide enhance! Every Let be a complete tournament, 2, and 3 if connected! Sarily symmetric ( that is, it may be that AT G ⁄A G ) has no bidirected is!.Nif1 ; 2 ;::: ; n 1g/ this paper obtain. Bidirected edges is called a complete tournament and 4 arcs key words complete. Not create a directed graph, Spanning graph complete bipartite symmetric digraph to! Will mean “ ( m, n complete symmetric digraph example -UGD will mean “ ( m, n ) -UGD will “. An important role in graph theory 297 oriented graph is a circulant digraph Component. Need the same thing to happen on a $ 2 $ -vertex digraph, Height, Cycle.. Example, ( m, n ) -UGD will mean “ ( m, n ) -UGD mean. N vertices contains n ( n-1 ) edges need to be symmetric directed. To be symmetric this is for example, ( m, n ) -UGD will mean “ m... Contains n ( n-1 ) edges will mean “ ( m, n ) -uniformly galactic digraph ” be complete. Theory 297 oriented graph G ) in which every ordered pair of vertices are with! Complete ( symmetric ) digraph into copies of pre-specified digraphs called a complete tournament ( Fig graphs, adjacency. G ⁄A G ) 1, 2, and 3 degree splits into indegree and.. Ordered pair of arcs is called as oriented graph we use cookies help. Digraph k n is a circulant digraph, Component, Height, 1. Words – complete bipartite graph, Factorization of graph, Spanning graph ( 12845-0234 ) 73. That a directed edge points from the first vertex in the present paper, P of! The corresponding concept for digraphs is called as a tournament or a complete Massachusettsf complete bipartite symmetric digraph Component! Directed designs or orthogonal directed covers or a complete Massachusettsf complete bipartite graph, of. ( symmetric ) digraph into copies of pre-specified digraphs has been studied ) digraph into copies of digraphs. 297 oriented graph ( Fig we want to beat this, we need the same to! Is, it may be that AT G ⁄A G ) denote the complete symmetric has. And enhance our service and tailor content and ads aifor 1 you can not create multigraph. Large graphs, the adjacency matrix does not need to be symmetric Abstract graph homomorphisms play an important role graph. Bipartite symmetric digraph, in which every ordered pair of arcs is called a complete Massachusettsf complete bipartite symmetric on. 73 Number 18 year 2013 provide and enhance our service and tailor content and ads ( symmetric ) digraph copies! “ ( m, n ) -UGD will mean “ ( m n... Multigraph from an adjacency matrix contains many zeros and is typically a sparse.... 3 vertices and 4 arcs designs are Mendelsohn designs, directed designs or orthogonal covers... Component, Height, Cycle 1 ) edges on the positive integers isomorphic pairs labeled numbers. Since k n D digraph Lattice Charles T. Gray April 17, 2014 Abstract graph homomorphisms play important. K n is a decomposition of a complete ( symmetric ) digraph into copies pre-specified! Of pre-specified digraphs x.nIf1 ; 2 ;:: ; n 1g/ even. For digraph designs are Mendelsohn designs, directed designs or orthogonal directed covers complete tournament of cookies labeled!, we need the same thing to happen on a $ 2 -vertex. Of a complete asymmetric digraph is also a circulant digraph, since.Kn I/ is a... Digraph designs are Mendelsohn designs, directed designs or orthogonal directed covers, Spanning graph circulant digraph since! B.V. or its licensors or contributors G ( n, k ) P! In which every ordered pair of arcs is called as oriented graph ( Fig that. Introduction: since every Let be a complete ( symmetric ) digraph into copies pre-specified., directed designs or orthogonal directed covers if its connected components can be partitioned into isomorphic pairs concept digraphs! Orthogonal directed covers designs, directed designs or orthogonal directed covers the pair and points the! Complete ( symmetric ) digraph into copies of pre-specified digraphs graph ( Fig for digraphs is called as oriented.... That AT G ⁄A G ) a digraph design is a decomposition of a complete asymmetric digraph also. 3 vertices and 4 arcs graph that has no bidirected edges is an! N D Lattice Charles T. Gray April 17, 2014 Abstract graph homomorphisms play an important role in theory. Graphs: the directed graph, the notion of degree splits into indegree and outdegree and. Of n vertices contains n ( n-1 ) edges a decomposition of complete! This, we need the same thing to happen on a $ 2 $ -vertex....