Raymond Greenlaw
- ALELIUNAS, R., Kaap, R. M., LIPTON, R. J., Lov~sz, L., AND RACKOFF, C. 1979. Random walks, umversal traversal sequences, and the complexity of maze problems. In 20th Annual Svrnposzurn on Foundations of Computer Sctence (San Juan, Puerto Rico, Oct.), IEEE, 218-223Google Scholar
- ALVAI, Y, CHARTRAND, G, OLLERMANN, O. R., AND SCHW~NK, A. J. Eds. 1991. Graph Theory, Cornblnatorlcs and Applzcat~ons. Wiley, New York.Google Scholar
- ANDERSON, R.J. 1985 The complexity of parallel algorithms. Stanford Univ, Ph.D. Thesis. Tech. Rep. STAN-CS-86-1092, Stanford Umv., Compurer Science Dept Google Scholar
- ANDERSON, R. J AND MAX-R, E. W. 1987. Parallelism and the maximal path problem Infi Process. Lett. 24, 2, 121-26. Google Scholar
- BAR-YEHUDA, R., AND EVEN S. 1985. A local-ratm theorem for approximating the weighted vertex cover problem. Ann. Dzscrete Math. 25, 27-46Google Scholar
- BERGE, C. 1972. Alternating chain methods A survey In Graph Theory and Computll~g, R. C. Read, Ed. Academic Press, New York.Google Scholar
- BERGE, C., Ed. 1973. Graphs and Hypergraphs. American Elsevier, New York.Google Scholar
- BIGGS, N. L., LLOYD, E. K., AND WILSON, R.J. 1976. Graph Theory 17,36-1936. Oxford University Press, London Google Scholar
- Bosd~K, J 1967. Hamfltoman lines m cubic graphs. In Theory of Graphs Internatzonal Syrnposturrt (International Computahon Cen- ~re, Rome l, Gordon and Breach, New York, 35-46Google Scholar
- BROOKS, R L. 1941 On colouring the nodes of a network. Proc. Carnbrtdge Phzt Soc 37, 194-197.Google Scholar
- Buss, J., AND TOMPA, M 1995 Lower bounds on universal sequences based on chains of length five. In Cornput. 120, 2 (Aug.), 326-329. Google Scholar
- CALAMONERI, T. 1992. Immersione nel piano di grafi cubici uno-retillinei. Univ. Rome, La Sapienza, Oct., Master's Thes~sGoogle Scholar
- CALAMONERI, T., AND PETRESCHI, R 1995. A parallel algorithm for orthogonal drawings of cubic graphs. Tecb Rep SI/RR 95/03, Univ of Rome, La Sapienza.Google Scholar
- CANTONI, Z. 1964 On cubic Hamilton graphs Ist Lornbardo sc. lett. rend. c. sc mat nat. 98, 319-326.Google Scholar
- CULIK, K. 1964. A remark to the construction of cubic graphs containing Hamiltonian lines. Casop. p~stov, mat. 89, 385-389. As cited in Graphs and Hypergraphs, 1973, C. Berge, Ed., American Elsevier, New York.Google Scholar
- COOK, S. A. 1971. The complexity of theorem proving procedures. In Thrrd Annual ACM Sympostum on Theory of Computing, (Shaker Heights, Ohio), 151-158. Google Scholar
- COPPERSMITH, D., AND WINOGRAD, S. 1990. Matrix multiplication via arithmetic progressions. J. Syrup. Comput. 9, 3, 251-280. Google Scholar
- ERDOS, P. AND SACHS, H. 1963. Regutiire Graphen gegebener Taillenweite mit mimmaler Knotenzahl. Tech. Rep. Math.-Nat. Reihe XII/3, Wiss. Z. Univ. Halle.Google Scholar
- EVEN, S. 1979. Graph Algorithms. Computer Science Press, Rockville, MD. Google Scholar
- EVEN, S., AND TARJAN, R.E. 1976. Computing an st-numbering. Theor. Comput. Scz. 2, 339-344.Google Scholar
- FREDERICKSON, G.N. 1985. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Comput. 14, 4, 781-798.Google Scholar
- FRIEZE, A., RADCLIFFE, A. J., AND SUEN S. 1993. Analysis of a simple greedy matching algorithm on random cubic graphs In the Fourth Annual ACM-SIAM Symposium on Dtscrete A1- gomthms (Austin, TX), ACM, New York, 341-351. Google Scholar
- GAREY, M. R., AND JOHNSON, D.S. 1979. Computers and intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York. Google Scholar
- GAREY, M. R, AND JOHNSON, D.S. 1977. The rectilinear Sterner tree problem is NP-complete. SIAM J. Appl. Math. 32, 4, 826-834.Google Scholar
- GAREY, M. R., GRAHAM, R. L., JOHNSON, D. S., AND KNUTH, D. E. 1978. Complexity results for bandwidth communication. SIAM J. Appl. Math. 34, 3, 447-495.Google Scholar
- GAREY, M. R., JOHNSON, D. S., AND STOCKMEYER, L. 1976a. Some simplified NP-complete graph problems. Theor. Comput. Scl. 1,237-267.Google Scholar
- GAREY, M. R., JOHNSON, D. S., AND TARJAN, R. E. 1976b. The planar Hamiltonian circuit problem is NP-comptete. SIAM J. Comput. 5, 4, 704-714.Google Scholar
- GAVRIL, F. 1977. Some NP-complete problems on graphs. In Proceedings of the Eleventh Conference on Informatzon Sctences and Systems (Baltimore, MD), 91-95.Google Scholar
- GREENLAW, R. 1994. Breadth-depth search is 9- complete. Parallel Process. Lett. 3, 3, 209-222.Google Scholar
- GREENLAW, R., HOOVER, H. J., AND Ruzzo, W. L. 1995. Limits to Parallel Computation: P-cornpleteness Theory. Oxford University Press, New York. Google Scholar
- HARARY, F. AND SCHWENK, A.J. 1972. Evolution of the path number of a graph: Covering and packing in graphs, II. In Graph Theory and Computing, R. C. Read, Ed., Academic Press, New York.Google Scholar
- ISRAELI, A., AND SHILOACH, Y. 1986. An improved algorithm for maximal matching. Inf Process. Lett. 33, 2, 57-60. Google Scholar
- IZBICKI, H. 1967. On the electronic generation of regular graphs. In Theory of Graphs International Sympostum (International Computation Centre, Rome), Gordon and Breach, New York, 177-182.Google Scholar
- JOHNSON, D. S. 1981. The NP-completeness column: An ongoing guide. J. Alg. 2, 4, 393-405. Google Scholar
- JOHNSON, E.L. 1963. A proof of the four-coloring of the edges of a regular three-degree graph. Tech. Rep. O. R. C. 63-28 (R. R.) Min. Report, Univ. of California, Operations Research Center. 1963. As cited in Graphs and Hypergraphs, 1973, C. Berge, Ed., American Elsevier, New York.Google Scholar
- KARP, R.M. 1972. Reducibility among combinatoriat problems. In Complexity of Computer Computations, R. E. Miller and J. W. Thatcher, Eds., Plenum Press, New York, 85-104.Google Scholar
- KARP, R. M., AND RAMACHANDRAN, V. 1990. Parallet algorithms for shared-memory machines. In Handbook of Theoretical Computer Sctence, vol. A, J. Leeuwan, Ed., MIT Press/Elsevier, Cambridge, Mass, Ch. 17, 869-941. Google Scholar
- KARTESZI, E. 1960. Piani finiti ciclici come resoluzioni dl un certo problema di minimo. Boll. Un. Mat. Ital. 15, 3, 522-528.Google Scholar
- KOBLER, J., SCHONING, U., AND TORfi, N, J. 1993. The Graph Isomorphism Problem: Its Structural Complexity. Birkhiiuser, Boston. Google Scholar
- KORFHAGE, R. Z. 1984. Discrete Computational Structures. 2nd ed. Academic Press, New York. Google Scholar
- KOTZIG, A. 1962. The construction of cubic Hamiltonian graphs. Casop. p6stov, mat. 87, 477-488; As cited in Graphs and Hypergraphs, C. Berge, Ed. 1973. American Elsevier, New York.Google Scholar
- KURATOWSKI, G. 1930. Sur le probt~me des courbes gauches en topologie. Fundamenta Math. 15, 271-283.Google Scholar
- LADNER, R.E. 1975. The circuit value problem is log space complete for P SIGACT News 7, 1 (Jan.), 18-20. Google Scholar
- LEIGHTON, F. T. 1992. Introductzon to Parallel Algorzthms and Architectures: Arrays, Trees, Hypercubes. Morgan-Kaufmann, San Mateo, Calif. Google Scholar
- LEMPEL, A., EVEN, S., AND CEDERBAUM, J. 1966. An algorithm for planarity testing of graphs. In Theory of Graphs, International Sympostum (Rome), 215-232.Google Scholar
- LIPSKY, W. 1977. Two NP-comptete problems related to information retrieval. In Fundamentals of Computatton Theory, Lecture Notes ~n Computer Sctence, vol. 56, Springer-Verlag, New York, 452-458.Google Scholar
- LIu, Y, MARCHIORO, P., AND PETRESCHI, R. 1994. At most single-bend embeddings of cubic graphs. Appl. Math. (Chinese journal) 9, 2, 127-142.Google Scholar
- LIU, Y., MARCHIORO, P., PETRESCHI, R., AND SIME- ONE, B. 1992. Theoretical results of at most l-bend embeddablhty of cubic graphs. Acta Math. Appl. Sznica, 8, 2, 188-192.Google Scholar
- LOEBL, M 1991. Efficient maximal cubic graph cuts. In Automata Languages and Programming: 18th International Colloquium (July, Madrid), Lecture Notes tn Computer Science, vot 510, Springer-Verlag, New York, 351-362 Google Scholar
- MAIER, D., AND STORER, J.A. 1977. A note on the complexity of the superstring problem. Tech Rep. Report No. 233, Princeton Umv As cited m Computers and IntractzabUzty" A Guzde to the Theory of NP-Completeness, M. R Garey, and D. S Johnson, Eds., W. H. Freeman, New York.Google Scholar
- McGEE, W. F 1960. A minimal cubic graph of girth seven. Canadzan Math. Bull. 3, 149-152.Google Scholar
- MILLER, G.L. 1977. Graph isomorphmm, general remarks. In Ninth Annual ACM Symposium on Theory of Computing (Boulder, Colo, May), 143-150. Google Scholar
- MIYANO, S 1989. The lexicographically first maximal subgraph problems. P-completeness and NC algorithms. Math. Syst. Theor. 22, 1, 47-73.Google Scholar
- MONIEN, B, AND SPECKENMEYER, E. 1983. Some further approximation algorithms for the vertex cover problem. In Trees ~n Algebra and Prograrnmtng, 8th Colloquium (Lille, France, March 1983), Lecture Notes zn Computer Sczence, Sprlnger-Verlag, New York, 341-349 Google Scholar
- MYNHARDT, C. 1991. On the difference between the domination and independent domination numbers of cubic graphs. In Graph Theory, Combtnatontcs and Applzcatzons, Alvai et al., Eds., Wdey, New York, 939-947Google Scholar
- NAOLE, J.F. 1967 A new method for combmatorial problems on graphs. In Theory of Graphs Internatzonal Symposium (Internatmnal Computation Centre, Rome), Gordon and Breach, New York, 257-262.Google Scholar
- ORE, O. 1967. The Four Color Problem. Academ~c Press, New YorkGoogle Scholar
- PAPADIMITRIOU, C I-{. 1976. The NP-completeness of the bandwidth minimization problem Computzng 16, 263-270Google Scholar
- PETERSEN, J. 1891. Die Theorie der regulaen Graphen. Acta Math. 15, 193-220.Google Scholar
- PREPARATA, F., AND VUILLEMIN, J. 1981. The cube-connected cycles: A versatile network for parallel computation Commun. ACM 24, 5, 300-309. Google Scholar
- READ, R. C., ED. 1972. Graph Theory and Comput~ng. Academic Press, New York.Google Scholar
- REIF, 01 H. 1985. Depth-first search is inherently sequential Inf. Process. Lett. 20, 5, 229-234Google Scholar
- SACHS, H. 1963a. On regular graphs with given girth. In Theory of Graphs and Its Applzcations, Proceeding of the Symposzum held in Smolenzce, Academic Press, 91-97.Google Scholar
- SACHS, H. 1963b. Regular graphs with given girth and restricted circmts. J. London Math. Soc. 38, 423-429.Google Scholar
- SACHS, H. 1967. Construction of non-Hamiltonian planar regular graphs of degrees 3, 4, and 5 with highest possible connectiwty. In Theory of Graphs Internatzonal Symposium (International Computation Centre, Rome), Gordon and Breach, New York, 373-388.Google Scholar
- SACHS, H. 1968. Em yon Kozyrev und Grunberg angegebener nicht-Hamiltomshcer kubischer planarer Graph. Beitra&ge zur Graphentheome, p. 127-130. As cited in Graphs and Hypergraphs, 1973, C Berge, Ed., American Elsewer, New York.Google Scholar
- SCHAEFER, T.J. 1978. The complexity ofsat~sfiability problems. In Proceedzngs of the Tenth Annual ACM Symposzum and the Theory of Computzng (New York, May) 216-226. Google Scholar
- SCHAFFER, A. A., AND YANNAKAKIS, M. 1991 Simple local search problems that are hard to solve. SIAM J Comput. 20, 1, (Feb.), 56-87. Google Scholar
- SOUSSE LIE R, R. 1963 Probl~mes plaisants et d~lectables Rev. Fr Rech. Opdrat~onnelle 29, 405-406Google Scholar
- TAIT, P. G 1878-1880. On the colourlng of maps. Proc. Royal Soc. Edtnborough 10, 501-503.Google Scholar
- TAIT, P. G. 1884. Listings topolog~e, Phtl. Mag. 17, 5, 30-46. Sczent~fzc Papers, Vol. II, 85-98. As cited in W. T. Tutte, Non-Hamittonian planat maps, m Graph Theory and Computing, 1972, R. C. Read, Ed., Academic Press, New York.Google Scholar
- TAaJAN, R. E. 1985 Amortized computational complexity. SIAM J. Alg. Dzscrete Meth 2, 6, 306-318.Google Scholar
- TOMPA, M. 1990 Lower bounds on universal traversal sequences for cycles and other lower degree graphs. SIAM J Comput. 21, 6 Google Scholar
- TUTTE, W. T. 1972 Non-Hamiltonian planar maps. In Graph Theory and Computzng 1972 R. C. Read, Ed, Academic Press, New York.Google Scholar
- TUTTE, W. T. 1960. A non-Hamdtoman planar graph. Acta Math. Acad. Sci. Hungar. 11, 371-375.Google Scholar
- TUTTE, W.T. 1947. A family of cub~caI graph5 Proc Cambmdge PhU. Soc. 43, 459-474Google Scholar
- TUTTE, W.T. 1946. On Hamdtoman circuits. J. London Math. Soc. 21, 98-101.Google Scholar
- vAN LEEUWAN, J Ed. 1990. Handbook of Theoret~cal Computer Sctence, volume A: Algorithms and Complexity MIT Press/Elsevier, Cambridge, MA. Google Scholar
- WALTHER, H. 1966. Uber das problem der exlstenz yon Hamdtonkre~sen in planaren, reguliiren graphen der grade 3, 4 und 5. Dlss. TH IlrnenauGoogle Scholar
- WALTHER, H. 1965. Ein kubischer, planarer, zyklisch fiinffach zusammenhiingender graph, der keinen Hamiltonkreis besitzt. Wiss. Z. TH II- menau 11, 163-166.Google Scholar
- WATKIN, J. J. AND WILSON, R.J. 1991. A survey of snarks. In Graph Theory, Combzl~atonics and Applications, Wiley, New York, 1129-1142.Google Scholar
- YANNAKAK~S, M. 1981. Edge-deletion problems. SIAM J. Comput. 10, 2, 297-309.Google Scholar
- YANNAKAKIS, M. aND GAVRIL, F. 1979. Edge dominating sets in graphs. Unpublished manuscript cited in Computers and Intract~btlity: A Guide to the Theory of NP-Completeness, M. R. Garey and D. S. Johnson, W. H. Freeman, New York.Google Scholar
- YANNAKAKIS, M. 1978. Node- and edge-deletion NP-complete problems. In Proceedtng,~ of the Tel~th Annual ACM Symposium on Theory of Computing (San Diego, CA, May) 253-264. Google Scholar
Index Terms
- Cubic graphs
Recommendations
The b -chromatic number and f -chromatic vertex number of regular graphs
The b -chromatic number of a graph G , denoted by b ( G ) , is the largest positive integer k such that there exists a proper coloring for G with k colors in which every color class contains at least one vertex adjacent to some vertex in each of the ...
Star colouring of bounded degree graphs and regular graphs
AbstractA k-star colouring of a graph G is a function f : V ( G ) → { 0 , 1 , … , k − 1 } such that f ( u ) ≠ f ( v ) for every edge uv of G, and every bicoloured connected subgraph of G is a star. The star chromatic number of G, χ s ( G ), is ...
Nowhere-Zero 5-Flows On Cubic Graphs with Oddness 4
Tutte's 5-flow conjecture from 1954 states that every bridgeless graph has a nowhere-zero 5-flow. It suffices to prove the conjecture for cyclically 6-edge-connected cubic graphs. We prove that every cyclically 6-edge-connected cubic graph with oddness ...
Reviews
Benjamin L. Schwartz
A cubic graph is an undirected, loop-free graph, every vertex of which is of degree 3. Cubic graphs arise in various graph theory problems, including Hamiltonian graphs, 4-coloring, matching, planar graphs, and network flows. This paper provides a catalog of the major known results on cubic graphs. Most theorems are not proven, but brief text descriptions of the proof ideas are included. For a few new results by the authors, full proofs are provided. Almost any reader will find new material here; I certainly did. The compilation of results is a valuable service to the field,<__?__Pub Caret> but the bibliography could be much more extensive. The authors cite a few early papers (though far too few), such as the venerable 1930 Kuratowski trailblazer, no longer of theoretical interest but fascinating for historical insight. On 4-colorability, neither Saaty's ill-timed but comprehensive book (just before the conjecture was proven) nor the proof itself, by Appel and Haken, is cited. Can you believe that the discussion of network flows leaves out Ford and Fulkerson__?__ For results, this paper is immensely worthwhile, but for sources, the survey remains to be done.
Access critical reviews of Computing literature here
Become a reviewer for Computing Reviews.
Comments