Uniformly resolvable decompositions of Kv into paths on two, three and four vertices

In this paper we consider uniformly resolvable decompositions of the complete graph K v , i.e.,?decompositions of K v whose blocks can be partitioned into factors and each factor contains pairwise isomorphic blocks. We determine necessary and sufficient conditions for the existence of a uniformly resolvable decomposition of K v into paths on two, three and four vertices.

[1]  Alan C. H. Ling,et al.  Maximum uniformly resolvable designs with block sizes 2 and 4 , 2009, Discret. Math..

[2]  Mario Gionfriddo,et al.  Uniformly resolvable H-designs with H = {P 3 ,P 4 } , 2014 .

[3]  Zsolt Tuza,et al.  Maximum uniformly resolvable decompositions of Kv and Kv-I into 3-stars and 3-cycles , 2013, Discret. Math..

[4]  Alexander Rosa,et al.  The Hamilton-Waterloo problem: the case of Hamilton cycles and triangle-factors , 2004, Discret. Math..

[5]  Gennian Ge,et al.  On uniformly resolvable designs with block sizes 3 and 4 , 2010, Des. Codes Cryptogr..

[6]  Giovanni Lo Faro,et al.  Resolvable 3-star designs , 2014, Discret. Math..

[7]  Scott A. Vanstone,et al.  Class-uniformly resolvable pairwise balanced designs with block sizes two and three , 1991, Discret. Math..

[8]  Peter Adams,et al.  On the Hamilton-Waterloo Problem , 2002, Graphs Comb..

[10]  W. L. Piotrowski,et al.  The solution of the bipartite analogue of the Oberwolfach problem , 1991, Discret. Math..

[11]  Joseph Douglas Horton.  Resolvable Path Designs , 1985, J. Comb. Theory, Ser. A.

[12]  Victor Scharaschkin,et al.  Complete solutions to the Oberwolfach problem for an infinite set of orders , 2009, J. Comb. Theory, Ser. B.

[13]  Don R. Lick,et al.  On λ-Fold Equipartite Oberwolfach Problem with Uniform Table Sizes , 2003 .

[14]  Charles J. Colbourn,et al.  Resolvable and Near Resolvable Designs , 1996 .

[15]  Douglas R. Stinson,et al.  On resolvable group-divisible designs with block size 3 , 1987 .

[16]  Jiuqiang Liu,et al.  The equipartite Oberwolfach problem with uniform tables , 2003, J. Comb. Theory, Ser. A.

[17]  Brett Stevens,et al.  The Hamilton–Waterloo problem for cycle sizes 3 and 4 , 2009 .

[18]  Mario Gionfriddo,et al.  On the existence of uniformly resolvable decompositions of Kv and Kv-I-I into paths and kites , 2013, Discret. Math..

[19]  Ernst Schuster.  Uniformly resolvable designs with index one and block sizes three and four - with three or five parallel classes of block size four , 2009, Discret. Math..

[20]  Claude Berge,et al.  Graphs and Hypergraphs , 2021, Clustering.

[21]  Rolf S. Rees,et al.  Uniformly resolvable pairwise balanced designs with blocksizes two and three , 1987, J. Comb. Theory, Ser. A.

[22]  Zsolt Tuza,et al.  Uniformly resolvable decompositions of Kv into P3 and K3 graphs , 2013, Discret. Math..

[23]  Tommaso Traetta,et al.  A complete solution to the two-table Oberwolfach problems , 2013, J. Comb. Theory, Ser. A.

[24]  Marco Buratti,et al.  On sharply vertex transitive 2-factorizations of the complete graph , 2005, J. Comb. Theory, Ser. A.

[25]  Mario Gionfriddo,et al.  Uniformly resolvable ℋ-designs with ℋ={P3, 4} , 2014, Australas. J Comb..

[26]  Darryn E. Bryant,et al.  On bipartite 2‐factorizations of kn − I and the Oberwolfach problem , 2011, J. Graph Theory.

[27]  Kazuhiko Ushio.  P3-Factorization of complete bipartite graphs , 1988, Discret. Math..

[28]  Alan C. H. Ling,et al.  The Hamilton—Waterloo problem: The case of triangle‐factors and one Hamilton cycle , 2009 .

[29]  Mario Gionfriddo,et al.  On uniformly resolvable {K2, Pk}-designs with with k=3, 4 , 2015, Contributions Discret. Math..

[30]  Ernst Schuster,et al.  Author's Personal Copy Discrete Mathematics , 2022 .

[31]  Jean-Claude Bermond,et al.  Existence of Resolvable Path Designs , 1990, Eur. J. Comb..

[32]  Douglas R. Stinson,et al.  The Oberwolfach problem and factors of uniform odd length cycles , 1989, J. Comb. Theory, Ser. A.

[33]  Salvatore Milici,et al.  A note on uniformly resolvable decompositions of Kv and Kv-I into 2-stars and 4-cycles , 2013, Australas. J Comb..

[34]  Ernst Schuster,et al.  Small Uniformly Resolvable Designs for Block Sizes 3 and 4 , 2013 .

[35]  William Pettersson,et al.  Bipartite 2-Factorizations of Complete Multipartite Graphs , 2015, J. Graph Theory.