Skip to main content Accessibility help
×
Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-19T22:57:59.085Z Has data issue: false hasContentIssue false

9 - Integer flows and orientations

Published online by Cambridge University Press:  05 May 2015

Hongjian Lai
Affiliation:
West Virginia University
Rong Luo
Affiliation:
West Virginia University
Cun-Quan Zhang
Affiliation:
West Virginia University
Lowell W. Beineke
Affiliation:
Purdue University, Indiana
Robin J. Wilson
Affiliation:
The Open University, Milton Keynes
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2015

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. N., Alon, N., Linial and R., Meshulam, Additive bases of vector spaces over prime fields, J. Combin. Theory (A) 57 (1991), 203–210.Google Scholar
2. K., Appel and W., Haken, Every map is four colorable, Part I: Discharging, Illinois J. Math. 21 (1977), 429–490.Google Scholar
3. K., Appel, W., Haken and J., Koch, Every map is four colorable, Part II: Reducibility, Illinois J. Math. 21 (1977), 491–567.Google Scholar
4. D., Archdeacon, Face coloring of embedded graphs, J. Graph Theory 8 (1984), 387–398.Google Scholar
5. J., Barát and C., Thomassen, Claw-decompositions and Tutte-orientations, J. Graph Theory 52 (2006), 135–146.Google Scholar
6. C., Berge, Graphs and Hypergraphs (transl. E., Minieka), North-Holland, 1973.Google Scholar
7. J. A., Bondy and U. S. R., Murty, Graph Theory with Applications, Macmillan, 1976.Google Scholar
8. P. A., Catlin, A reduction method to find spanning eulerian subgraph, J. Graph Theory 12 (1988), 29–45.Google Scholar
9. P. A., Catlin, Double cycle covers and the Petersen graph, J. Graph Theory 13 (1989), 465–483.Google Scholar
10. P. A., Catlin, Supereulerian graphs: a survey, J. Graph Theory 16 (1992), 177–196.Google Scholar
11. U. A., Celmins, On Cubic Graphs that do not have an Edge 3-coloring, Ph.D. thesis, University of Waterloo (1984).Google Scholar
12. Z.-H., Chen and H.-J., Lai, Reductions techniques for supereulerian graphs and related topics – a survey, Combinatorics and Graph Theory 95, Vol. 1 (ed. Ku, Tung-Hsin), World Scientific (1995), 53–69.Google Scholar
13. M., DeVos, L., Goddyn, B., Mohar, D., Vertigan and X.-D., Zhu, Coloring-flow duality of embedded graphs, Trans. Amer. Math. Soc. 357 (2005), 3993–4016.Google Scholar
14. M., DeVos, J., Nešetřil and A., Raspaud, On edge-maps whose inverse preserves flows and tensions, Graph Theory, Trends in Mathematics (eds. J. A., Bondy, J., Fonlupt, J.-L., Fouquet, J.-C., Fournier and J. L. R., Alfonsin), Birkhäuser (2006), 109–138.Google Scholar
15. M., DeVos, R., Xu and G., Yu, Nowhere-zero Z3-connectivity, Discrete Math. 306 (2006), 26–30.Google Scholar
16. G.-H., Fan, H.-J., Lai, R., Xu, C.-Q., Zhang and C., Zhou, Nowhere-zero 3-flows in triangularly connected graphs, J. Combin. Theory (B) 98 (2008), 1325–1336.Google Scholar
17. H., Fleischner, Bipartizing matchings and Sabidussi's compatibility conjecture, Discrete Math. 244 (2002), 77–82.Google Scholar
18. J. L., Fouquet, Conjecture du 5-flot pour les graphes presque planaires, Sèminaire de Mathematiques Discrètes et Applications, Grenoble, 7 November 1985.Google Scholar
19. A., Galluccio, L. A., Goddyn and P., Hell, Large girth graphs avoiding a fixed minor are nearly bipartite, J. Combin. Theory (B) 83 (2001), 1–14.Google Scholar
20. L. A., Goddyn, M., Tarsi and C.-Q., Zhang, On (k, d)-colorings and fractional nowhere zero flows, J. Graph Theory 28 (1998), 155–161.Google Scholar
21. H., Grötzsch, Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel, Wiss. Z. der Martin-Luther-Universitat Halle-Wittenberg, Math.-Naturwissenschaftliche Reihe 8 (1958/59), 109–120.Google Scholar
22. B., Grünbaum, Grötzsch's theorem on 3-colorings, Michigan Math. J. 10 (1963), 303–310.Google Scholar
23. P. J., Heawood, Map color theorem, Quart. J. Pure and Applied Math. 24 (1890), 332–338.Google Scholar
24. W., Imrich, I., Peterin, S., Špacapan and C.-Q., Zhang, NZ-flows in strong products of graphs, J. Graph Theory 64 (2010), 267–276.Google Scholar
25. F., Jaeger, On interval hypergraphs and nowhere-zero flow in graphs, Research Report of Mathematics Application and Information, Uriiversité Scientifique et Medicale et Institut National Polytechnique de Grenoble 126, 1978.Google Scholar
26. F., Jaeger, Flows and generalized coloring theorems in graphs, J. Combin. Theory (B) 26 (1979), 205–216.Google Scholar
27. F., Jaeger, On circular flows in graphs, Finite and Infinite Sets (Eger, 1981), Colloq. Math. Soc. Janos Bolyai 37, North-Holland (1984), 391–402.Google Scholar
28. F., Jaeger, Nowhere-zero flow problems, Selected Topics in Graph Theory 3,(eds. L.W., Beineke and R. J., Wilson), Academic Press (1988), 71–95.Google Scholar
29. F., Jaeger, N., Linial, C., Payan and M., Tarsi, Group connectivity of graphs – A nonho-mogeneous analogue of nowhere-zero flow properties, J. Combin. Theory (B) 56 (1992), 165–182.Google Scholar
30. P. A., Kilpatrick, Tutte's First Colour-Cycle Conjecture, Ph.D. thesis, Cape Town, 1975.Google Scholar
31. M., Kochol, Cubic graphs without a Petersen minor have nowhere-zero 5-flows, Acta Math. Univ. Comenianae LXVIII (1999), 249–252.Google Scholar
32. M., Kochol, An equivalent version of the 3-flow conjecture, Discrete Math. 83 (2001), 258–261.Google Scholar
33. M., Kochol, Reduction of the 5-flow conjecture to cyclically 6-edge-connected snarks, J. Combin. Theory (B) 90 (2004), 139–145.Google Scholar
34. M., Kochol, Smallest counterexample to the 5-flow conjecture has girth at least eleven, J. Combin. Theory (B) 100 (2010), 381–389.Google Scholar
35. H.-J., Lai, Extending a partial nowhere zero 4-flow, J. Graph Theory 30 (1999), 277–288.Google Scholar
36. H.-J., Lai, Nowhere-zero 3-flows in locally connected graphs. J. Graph Theory 42 (2003), 211–219.Google Scholar
37. H.-J., Lai, Mod (2p + 1)-orientations and K2p+1-decompositions, SIAM J. Discrete Math. 21 (2007), 844–850.Google Scholar
38. H.-J., Lai and X.-W., Li, Group Chromatic number of planar graphs with girth at least 4, J. Graph Theory 52 (2006), 51–72.Google Scholar
39. H.-J., Lai and C.-Q., Zhang, Nowhere-zero 3-flows of highly connected graphs, Discrete Math. 110 (1992), 179–183.Google Scholar
40. C. H. C., Little, W. T., Tutte and D. H., Younger, A theorem on integer flows, Ars Combin. 26A (1988), 109–112.Google Scholar
41. L. M., Lovász, C., Thomassen, Y.-Z., Wu and C.-Q., Zhang, Nowhere-zero 3-flows and modulo k-orientations, J. Combin. Theory (B) 103 (2013), 587–598.Google Scholar
42. M., MÖller, H. G., Carstens and G., Brinkmann, Nowhere-zero flows in low genus graphs, J. Graph Theory 12 (1988), 183–190.Google Scholar
43. C. St.J. A, Nash-Williams, Edge-disjoint spanning trees of finite graphs, J. London Math. Soc. 36 (1961), 445–450.Google Scholar
44. Z., Pan and X.-D., Zhu, Construction of graphs with given circular flow numbers. J. Graph Theory 43 (2003), 304–318.Google Scholar
45. M., Preissmann, Surles Colorations des Aretes des Graphes Cubiques, Thèse de Doctorat de 3eme, Université de Grenoble, 1981.Google Scholar
46. N., Robertson, P. D., Seymour and R., Thomas, Girth six cubic graphs have Petersen minors, arXiv:1405.0533.
47. P. D., Seymour, Nowhere-zero 6-flows, J. Combin. Theory (B) 30 (1981), 130–135.Google Scholar
48. P. D., Seymour, Tutte's three-edge-colouring conjecture, Proceedings of Graph Theory@Georgia Tech, a conference honouring the 50th birthday of Robin Thomas, May 2012, https://smartech.gatech.edu/handle/1853/44224.
49. E., Steffen, Tutte's 5-flow conjecture for highly cyclically connected cubic graphs, Discrete Math. 310 (2010), 385–389.Google Scholar
50. E., Steffen, Intersecting 1-factors and nowhere-zero 5-flows, preprint, 2012.
51. R., Steinberg and D. H., Younger, Grötzsch's theorem for the projective plane, Ars Combin. 28 (1989), 15–31.Google Scholar
52. R., Thomas, Generalizations of the four color theorem, http://people.math.gatech.edu/~thomas/FC/generalize.html.
53. C., Thomassen, Grötzsch's 3-color theorem and its counterparts for the torus and the projective plane, J. Combin. Theory (B) 62 (1994), 268–279.Google Scholar
54. C., Thomassen, The weak 3-flow conjecture and the weak circular flow conjecture, J. Combin. Theory (B) 102 (2012), 521–529.Google Scholar
55. W. T., Tutte, On the imbedding of linear graphs in surfaces, Proc. London Math. Soc. (2) 51 (1949), 474–483.Google Scholar
56. W. T., Tutte, A contribution on the theory of chromatic polynomial, Canad. J. Math. 6 (1954), 80–91.Google Scholar
57. W. T., Tutte, A class of Abelian groups, Canad. J. Math. 8 (1956), 13–28.Google Scholar
58. W. T., Tutte, On the problem of decompositing a graph into n connected factors, J. London Math. Soc. 36 (1961), 221–230.Google Scholar
59. W. T., Tutte, On the algebraic theory of graph colourings, J. Combin. Theory 1 (1966), 15–50.Google Scholar
60. W. T., Tutte, A geometrical version of the four color problem, Combinatorial Mathematics and its Applications (eds. R. C., Bose and T. A., Dowling), Univ. North Carolina Press, 1967.Google Scholar
61. R., Xu and C.-Q., Zhang, Nowhere-zero 3-flows in squares of graphs, Electron. J. Combin. 10 (2003), R5.Google Scholar
62. D. H., Younger, Integer flows, J. Graph Theory 7 (1983), 349–357.Google Scholar
63. C.-Q., Zhang, Integer Flows and Cycle Covers of Graphs, Marcel Dekker, 1997.Google Scholar
64. C.-Q., Zhang, Circular flows of nearly eulerian graphs and vertex-splitting, J. Graph Theory 40 (2002), 147–161.Google Scholar
65. X. D., Zhu, Recent developments in circular colouring of graphs, Topics in Discrete Mathematics (eds. M., Klazar et al.), Springer (2006), 497–550.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×