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Isomorphic factorizations. X: Unsolved problems. (English) Zbl 0617.05051

The discussion of results and unsolved problems on isomorphic factorizations is divided into the following sections: 1. Complete graphs, 2. Combinatorial designs, 3. Complete multipartite graphs, 4. Isomorphic Ramsey numbers, 5. Directed graphs, 6. Automorphisms, 7. Regular graphs and tournaments, 8. Bisectable trees, 9. Even trees, 10. Unsolved problems (containing problems related to those discussed above).
Reviewer: St.Znám

MSC:

05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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[1] Alon, Ars Combin. 12 pp 117– (1981)
[2] Alon, Ars Combin. 14 pp 123– (1982)
[3] Alspach, J. Graph Theory 6 pp 441– (1982)
[4] Alspach, J. Graph Theory 9 pp 177– (1985)
[5] Anderson, Discrete Math. 18 pp 227– (1977)
[6] Anderson, Aequationes Math. 15 pp 201– (1977)
[7] Aubert, Discrete Math. 37 pp 19– (1981)
[8] Aubert, J. Combin. Theory Ser. B 32 pp 347– (1982)
[9] Bermond, Ars. Combin. 5 pp 293– (1978)
[10] Bermond, Congr. Numer. 15 pp 53– (1976)
[11] and , Cycle and circuit designs: Odd case. Beiträge zur Graphentheorie und deren Anwendungen. Technische Hochschule Ilmenau (1977) 11–32.
[12] Bialostocki, Acta Math. Acad. Sci. Hungar. 40 pp 201– (1982)
[13] Bloom, Ann. N.Y. Acad. Sci. 328 pp 32– (1979)
[14] Generalized ramsey theory for graphs–A survey. Graphs and Combinatorics, Springer Lecture Notes in Math. 406, Springer, Berlin (1974) 52–75.
[15] Caccetta, Congr. Numer. 23 pp 217– (1979)
[16] Parallelisms of Complete Designs. Cambridge Univ. Press, Cambridge (1976). · Zbl 0333.05007 · doi:10.1017/CBO9780511662102
[17] The decomposition of graphs into graphs having two edges, unpublished.
[18] Caro, Ars Combin. 9 pp 119– (1980)
[19] Chetwynd, Regular graphs of high degree are 1-factorizable · Zbl 0561.05027
[20] and , Recent results in graph decompositions. Combinatorics, London Math. Soc. Lecture Note Series 52, Cambridge (1981) 103–123. · doi:10.1017/CBO9780511662157.007
[21] Clapham, Discrete Math. 8 pp 251– (1974)
[22] Gelling, Congr. Numer. 9 pp 213– (1974)
[23] Golomb, Amer. Math. Monthly 88 pp 680– (1981)
[24] Graham, Isomorphic factorizations IX: Even trees
[25] Graph Theory, Addison-Wesley, Reading MA (1969).
[26] Harary, Ann. N.Y. Acad. Sci. 175 pp 198– (1970) · Zbl 0226.05119 · doi:10.1111/j.1749-6632.1970.tb56470.x
[27] , and , Decomposition of complete symmetric digraphs into the four oriented quadrilaterals, Combinatorial Mathematics, Springer Lecture Notes in Math. 686, Springer, Berlin (1978) 165–173.
[28] Harary, Pacific J. Math. 80 pp 435– (1979) · Zbl 0385.05049 · doi:10.2140/pjm.1979.80.435
[29] Harary, Combinatorica
[30] Harary, Amer. Math. Monthly
[31] Harary, Trans. Amer. Math. Soc. 242 pp 243– (1978)
[32] , and , Isomorphic factorisations III: Complete multipartite graphs, Combinatorial Mathematics, Springer Lecture Notes in Math. 686. Springer, Berlin (1978) 47–54.
[33] Harary, Mathematika 25 pp 279– (1978)
[34] Harary, Congr. Numer. 19 pp 13– (1978)
[35] Hell, Aequationes Math. 12 pp 1– (1975)
[36] Himelwright, J. Graph Theory 6 pp 75– (1982)
[37] J. Graph Theory 8 pp 185– (1984)
[38] Himelwright, Elem. Math. 29 pp 6– (1974)
[39] and , Isomorphism of planar graphs. Complexity of Computer Computations ( and , eds.), Plenum, New York (1972) 131–152. · doi:10.1007/978-1-4684-2001-2_13
[40] Horton, Ann. Discrete Math.
[41] Huang, Utilitas Math. 10 pp 77– (1976)
[42] Huang, Ars Combin. 5 pp 23– (1978)
[43] Hung, J. Combin. Theory Ser. A 14 pp 310– (1973)
[44] Köhler, Über das Oberwolfacher Problem · Zbl 0384.05015
[45] Kotzig, Utilitas Math. 4 pp 261– (1973)
[46] Kotzig, Congr. Numer. 21 pp 457– (1978)
[47] Kotzig, J. Combin. Theory Ser. B 31 pp 292– (1981)
[48] Kotzig, Bull. London Math. Soc. 7 pp 51– (1975)
[49] Lindner, J. Combin. Theory Ser. B 20 pp 265– (1976)
[50] Mahmoodian, Canad. Math. Bull. 24 pp 107– (1981) · Zbl 0473.05030 · doi:10.4153/CMB-1981-017-9
[51] Mendelsohn, J. Combin. Theory Ser. A 25 pp 97– (1978)
[52] Topics on Tournaments, Holt, Rinehart and Winston, New York (1968).
[53] Factorisation of complete bipartite graphs into two isomorphic subgraphs, Combinatorial Mathematics VI, Springer Lecture Notes in Math. 748, Springer, Berlin (1979) 98–111.
[54] Quinn, J. Graph Theory 7 pp 285– (1983)
[55] On the Existence and Enumeration of Isomorphic Factorisations and Graphs, Master’s Thesis, University of Newcastle, Australia (1984).
[56] Rao, J. Combin. Theory Ser. B 22 pp 1– (1977)
[57] and , Solution of Kirkman’s schoolgirl problem. Combinatorics (Proc. Sympos. Pure Math. vol. 19), Amer. Math. Soc., Providence (1971) 187–203. · Zbl 0248.05009 · doi:10.1090/pspum/019/9959
[58] Read, J. London Math. Soc. 38 pp 99– (1963)
[59] Ringel, Arch. Math. 14 pp 354– (1963)
[60] Problem 25, Theory of Graphs and Its Applications (ed.), Academic, New York (1964) 162.
[61] Isomorphic factorisations VI: Automorphisms, Combinatorial Mathematics VI, Springer Lecture Notes in Math. 748, Springer, Berlin (1979) 127–136.
[62] Rosa, Discrete Math. 12 pp 269– (1975)
[63] Rosa, Discrete Appl. Math. 4 pp 291– (1982)
[64] On Isomorphic Decompositions of Graphs, Ph.D. thesis, Western Michigan University, Kalamazoo (1983).
[65] Combinatorial Mathematics, Carus Monograph 14, Math. Assoc. Amer. Wiley, New York (1963).
[66] Sachs, Publ. Math. Debrecen 9 pp 270– (1962)
[67] Tarsi, Discrete Math. 26 pp 273– (1979)
[68] Ushio, Hiroshima Math. J. 8 pp 207– (1978)
[69] Wallis, J. Combin. Inform. Syst. Sci. 2 pp 104– (1977)
[70] One-factorisations of wreath products, Combinatorial Mathematics VIII, Springer Lecture Notes in Math. 884 (1981) 337–345. · doi:10.1007/BFb0091829
[71] , and , Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices, Springer Lecture Notes in Math. 292, Springer, Berlin (1972). · doi:10.1007/BFb0069907
[72] Wang, Scientia Sinica Ser. A 25 pp 1152– (1982)
[73] Wang, Scientia Sinica Ser. A 26 pp 1167– (1983)
[74] Construction and uses of pairwise balanced designs. Combinatorics ( and , eds.), Math. Centre Tracts 55, Math. Centrum, Amsterdam (1974) 18–41.
[75] Wilson, Congr. Numer. 15 pp 647– (1976)
[76] Wormald, J. Graph Theory 8 pp 117– (1984)
[77] Yamamoto, Hiroshima Math. J. 5 pp 33– (1975)
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