Independent Sets and Matchings

  • R. Balakrishnan
  • K. Ranganathan
Chapter
Part of the Universitext book series (UTX)

Abstract

Vertex-independent sets and vertex coverings as also edge-independent sets and edge coverings of graphs occur very naturally in many practical situations and hence have several potential applications. In this chapter, we study the properties of these sets. In addition, we discuss matchings in graphs and, in particular, in bipartite graphs. Matchings in bipartite graphs have varied applications in operations research. We also present two celebrated theorems of graph theory, namely, Tutte’s 1-factor theorem and Hall’s matching theorem. All graphs considered in this chapter are loopless.

References

  1. 1.
    Adiga, C., Balakrishnan, R., So, W.: The skew energy of a digraph. Linear Algebra Appl. 432, 1825–1835 (2010)MathSciNetMATHGoogle Scholar
  2. 2.
    Aharoni, R., Szabó, T.: Vizing’s conjecture for chordal graphs. Discrete Math. 309(6), 1766–1768 (2009)MathSciNetMATHGoogle Scholar
  3. 3.
    Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, MA (1974)MATHGoogle Scholar
  4. 4.
    Akbari, S., Moazami, F., Zare, S.: Kneser graphs and their complements are hyperenergetic. MATCH Commun. Math. Comput. Chem. 61, 361–368 (2009)MathSciNetMATHGoogle Scholar
  5. 5.
    Apostol, T.M.: Introduction to Analytic Number Theory. Springer-Verlag, Berlin (1989)Google Scholar
  6. 6.
    Appel, K., Haken, W.: Every planar map is four colorable: Part I-discharging. Illinois J. Math. 21, 429–490 (1977)MathSciNetMATHGoogle Scholar
  7. 7.
    Appel, K., Haken, W.: The solution of the four-color-map problem, Sci. Amer. 237(4) 108–121 (1977)MathSciNetGoogle Scholar
  8. 8.
    Appel, K., Haken, W., Koch, J.: Every planar map is four colorable: Part II—reducibility. Illinois J. Math. 21, 491–567 (1977)MathSciNetMATHGoogle Scholar
  9. 9.
    Aravamudhan, R., Rajendran, B.: Personal communicationGoogle Scholar
  10. 10.
    Balakrishnan, R.: The energy of a graph. Linear Algebra Appl. 387, 287–295 (2004)MathSciNetMATHGoogle Scholar
  11. 11.
    Balakrishnan, R., Paulraja, P.: Powers of chordal graphs. J. Austral. Math. Soc. Ser. A 35, 211–217 (1983)MathSciNetMATHGoogle Scholar
  12. 12.
    Balakrishnan, R., Paulraja, P.: Chordal graphs and some of their derived graphs. Congressus Numerantium 53, 71–74 (1986)MathSciNetGoogle Scholar
  13. 13.
    Balakrishnan, R., Kavaskar, T., So, W.: The energy of the Mycielskian of a regular graph. Australasian J. Combin. 52, 163–171 (2012)MathSciNetMATHGoogle Scholar
  14. 14.
    Bapat, R.B.: Graphs and Matrices, Universitext Series, Springer. (2011)MATHGoogle Scholar
  15. 15.
    Barcalkin, A.M., German, L.F.: The external stability number of the Cartesian product of graphs. Bul. Akad. Štiince RSS Moldoven 94(1), 5–8 (1979)MathSciNetGoogle Scholar
  16. 16.
    Behzad, M., Chartrand, G., Lesniak-Foster, L.: Graphs and Digraphs. Prindle, Weber & Schmidt International Series, Boston, MA (1979)Google Scholar
  17. 17.
    Beineke, L.W.: On derived graphs and digraphs. In: Sachs, H., Voss, H.J., Walther, H. (eds.) Beiträge zur Graphentheorie, pp. 17–33. Teubner, Leipzig (1968)Google Scholar
  18. 18.
    Benzer, S.: On the topology of the genetic fine structure. Proc. Nat. Acad. Sci. USA 45, 1607–1620 (1959)Google Scholar
  19. 19.
    Berge, C.: Graphs and Hypergraphs, North-Holland Mathematical Library, Elsevier, 6, (1973)Google Scholar
  20. 20.
    Berge, C.: Graphes, Third Edition. Dunod, Paris, 1983 (English, Second and revised edition of part 1 of the 1973 English version, North-Holland, 1985)MATHGoogle Scholar
  21. 21.
    Berge, C., Chvátal, V.: Topics on perfect graphs. Annals of Discrete Mathematics, 21. North Holland, Amsterdam (1984)Google Scholar
  22. 22.
    Biggs, N.: Algebraic Graph Theory, 2nd ed. Cambridge University Press, Cambridge (1993)Google Scholar
  23. 23.
    Birkhoff, G., Lewis, D.: Chromatic polynomials. Trans. Amer. Math. Soc. 60, 355–451 (1946)MathSciNetMATHGoogle Scholar
  24. 24.
    Bondy, J.A.: Pancyclic graphs. J. Combin. Theory Ser. B 11, 80–84 (1971)MathSciNetMATHGoogle Scholar
  25. 25.
    Bondy, J.A., Chvátal, V.: A method in graph theory. Discrete Math. 15, 111–135 (1976)MathSciNetMATHGoogle Scholar
  26. 26.
    Bondy, J.A., Halberstam, F.Y.: Parity theorems for paths and cycles in graphs. J. Graph Theory 10, 107–115 (1986)MathSciNetMATHGoogle Scholar
  27. 27.
    Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. MacMillan, India (1976)MATHGoogle Scholar
  28. 28.
    Bollobás, B.: Modern Graph Theory, Graduate Texts in Mathematics. Springer (1998)MATHGoogle Scholar
  29. 29.
    Brešar, B., Rall, D.F.: Fair reception and Vizing’s conjecture. J. Graph Theory 61(1), 45–54 (2009)MathSciNetMATHGoogle Scholar
  30. 30.
    Brešsar, B., Dorbec, P., Goddard, W., Hartnell, B.L., Henning, M.A., Klavžar, S., Rall, D.F.: Vizing’s conjecture: A survey and recent results. J. Graph Theory, 69, 46–76 (2012)MathSciNetGoogle Scholar
  31. 31.
    Brooks, R.L.: On coloring the nodes of a network. Proc. Cambridge Philos. Soc. 37, 194–197 (1941)MathSciNetGoogle Scholar
  32. 32.
    Bryant, D.: Another quick proof that K 10P + P + P. Bull. ICA Appl. 34, 86 (2002)Google Scholar
  33. 33.
    Cayley, A.: On the theory of analytical forms called trees. Philos. Mag. 13, 172–176 (1857); Mathematical Papers, Cambridge 3, 242–246 (1891)Google Scholar
  34. 34.
    Chartrand, G., Ollermann, O.R.: Applied and algorithmic graph theory. International Series in Pure and Applied Mathematics. McGraw-Hill, New York (1993)Google Scholar
  35. 35.
    Chartrand, G., Wall, C.E.: On the Hamiltonian index of a graph. Studia Sci. Math. Hungar. 8, 43–48 (1973)MathSciNetGoogle Scholar
  36. 36.
    Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006)MathSciNetMATHGoogle Scholar
  37. 37.
    Chung, F.R.K.: Diameters and eigenvalues. J. Amer. Math. Soc. 2, 187–196 (1989)MathSciNetMATHGoogle Scholar
  38. 38.
    Chvátal, V.: On Hamilton’s ideals. J. Combin. Theory Ser. B 12, 163–168 (1972)MATHGoogle Scholar
  39. 39.
    Chvátal, V., Erdős, P.: A note on Hamiltonian circuits. Discrete Math. 2, 111–113 (1972)MathSciNetMATHGoogle Scholar
  40. 40.
    Chvátal, V., Sbihi, N.: Bull-free Berge graphs are perfect. Graphs Combin. 3, 127–139 (1987)MathSciNetMATHGoogle Scholar
  41. 41.
    Clark, J., Holton, D.A.: A First Look at Graph Theory. World Scientific, Teaneck, NJ (1991)MATHGoogle Scholar
  42. 42.
    Clark, W.E., Suen, S.: An inequality related to Vizings conjecture. Electron. J. Combin. 7(1), Note 4, (electronic) (2000)Google Scholar
  43. 43.
    Clark, W.E., Ismail, M.E.H., Suen, S.: Application of upper and lower bounds for the domination number to Vizing’s conjecture. Ars Combin. 69, 97–108 (2003)MathSciNetMATHGoogle Scholar
  44. 44.
    Cockayne, E.J.: Domination of undirected graphs—a survey, In: Alavi, Y., Lick, D.R. (eds.) Theory and Application of Graphs in America’s Bicentennial Year. Springer-Verlag, New York (1978)Google Scholar
  45. 45.
    Cockayne, E.J., Hedetniemi, S.T., Miller, D.J.: Properties of hereditary hypergraphs and middle graphs. Networks 7, 247–261 (1977)MathSciNetMATHGoogle Scholar
  46. 46.
    Cvetković, D.M.: Graphs and their spectra. Thesis, Univ. Beograd Publ. Elektrotehn. Fak., Ser. Mat. Fiz., No. 354–356, pp. 01–50 (1971)Google Scholar
  47. 47.
    Cvetković, D.M., Doob, M., Sachs, H.: Spectra of Graphs—Theory and Application, Third revised and enlarged edition. Johann Ambrosius Barth Verlag, Heidelberg/Leipzig (1995)Google Scholar
  48. 48.
    Deligne, P.: La conjecture de Weil I. Publ. Math. IHES 43, 273–307 (1974)MathSciNetGoogle Scholar
  49. 49.
    Demoucron, G., Malgrange, Y., Pertuiset, R.: Graphes planaires: Reconnaissance et construction de représentaitons planaires topologiques. Rev. Française Recherche Opérationnelle 8, 33–47 (1964)Google Scholar
  50. 50.
    Descartes, B.: Solution to advanced problem no. 4526. Am. Math. Mon. 61, 269–271 (1954)Google Scholar
  51. 51.
    Diestel, R.: Graph theory. In: Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2010)Google Scholar
  52. 52.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Math. 1, 269–271 (1959)MathSciNetMATHGoogle Scholar
  53. 53.
    Dirac, G.A.: A property of 4-chromatic graphs and some remarks on critical graphs. J. London Math. Soc. 27, 85–92 (1952)MathSciNetMATHGoogle Scholar
  54. 54.
    Dirac, G.A.: Some theorems on abstract graphs. Proc. London Math. Soc. 2, 69–81 (1952)MathSciNetMATHGoogle Scholar
  55. 55.
    Dirac, G.A.: Généralisations du théoréme de Menger. C.R. Acad. Sci. Paris 250, 4252–4253 (1960)MathSciNetMATHGoogle Scholar
  56. 56.
    Dirac, G.A.: On rigid circuit graphs-cut sets-coloring. Abh. Math. Sem. Univ. Hamburg 25, 71–76 (1961)MathSciNetMATHGoogle Scholar
  57. 57.
    Drinfeld, V.: The proof of Peterson’s conjecture for GL(2) over a global field of characteristic p. Funct. Anal. Appl. 22, 28–43 (1988)Google Scholar
  58. 58.
    Droll, A.: A classification of Ramanujan Cayley graphs. Electron. J. Cominator. 17, #N29 (2010)Google Scholar
  59. 59.
    El-Zahar, M., Pareek, C.M.: Domination number of products of graphs. Ars Combin. 31 (1991) 223–227MathSciNetMATHGoogle Scholar
  60. 60.
    Fáry, I.: On straight line representation of planar graphs. Acta Sci. Math. Szeged 11, 229–233 (1948)MATHGoogle Scholar
  61. 61.
    Ferrar, W.L.: A Text-Book of Determinants, Matrices and Algebraic Forms. Oxford University Press (1953)Google Scholar
  62. 62.
    Fiorini, S., Wilson, R.J.: Edge-colourings of graphs. Research Notes in Mathematics, vol. 16. Pitman, London (1971)Google Scholar
  63. 63.
    Fleischner, H.: Elementary proofs of (relatively) recent characterizations of Eulerian graphs. Discrete Appl. Math. 24, 115–119 (1989)MathSciNetMATHGoogle Scholar
  64. 64.
    Fleischner, H.: Eulerian graphs and realted topics. Ann. Disc. Math. 45 (1990)Google Scholar
  65. 65.
    Ford, L.R. Jr., Fulkerson, D.R.: Maximal flow through a network. Canad. J. Math. 8, 399–404 (1956)MathSciNetMATHGoogle Scholar
  66. 66.
    Ford, L.R. Jr., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton (1962)MATHGoogle Scholar
  67. 67.
    Fournier, J.-C.: Colorations des arétes d’un graphe. Cahiers du CERO 15, 311–314 (1973)MathSciNetMATHGoogle Scholar
  68. 68.
    Fournier, J.-C.: Demonstration simple du theoreme de Kuratowski et de sa forme duale. Discrete Math. 31, 329–332 (1980)MathSciNetMATHGoogle Scholar
  69. 69.
    Fulkerson, D.R.: Blocking and anti-blocking pairs of polyhedra. Math. Programming 1, 168–194 (1971)MathSciNetMATHGoogle Scholar
  70. 70.
    Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pacific J. Math. 15, 835–855 (1965)MathSciNetMATHGoogle Scholar
  71. 71.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman & Co., San Francisco (1979)MATHGoogle Scholar
  72. 72.
    Gibbons, A.: Algorithmic Graph Theory. Cambridge University Press, Cambridge (1985)MATHGoogle Scholar
  73. 73.
    Gilmore, P.C., Hoffman, A.J.: A characterization of comparability graphs and interval graphs. Canad. J. Math. 16, 539–548 (1964)MathSciNetMATHGoogle Scholar
  74. 74.
    Goddard, W.D., Kubicki, G., Ollermann, O.R., Tian, S.L.: On multipartite tournaments. J. Combin. Theory Ser. B 52, 284–300 (1991)MathSciNetMATHGoogle Scholar
  75. 75.
    Godsil, C.D., Royle, G.: Algebraic graph theory. Graduate Texts in Mathematics, vol. 207. Springer-Verlag, Berlin (2001)Google Scholar
  76. 76.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)MATHGoogle Scholar
  77. 77.
    Gould, R.J.: Graph Theory. Benjamin/Cummings Publishing Company, Menlo Park, CA (1988)MATHGoogle Scholar
  78. 78.
    Grinberg, É.Ja.: Plane homogeneous graphs of degree three without Hamiltonian circuits (Russian). Latvian Math. Yearbook 4, 51–58 (1968)Google Scholar
  79. 79.
    Gross, J.L., Tucker, T.W.: Topological Graph Theory. Wiley, New York (1987)MATHGoogle Scholar
  80. 80.
    Gross, J.L., Yellen, J.: Handbook of graph theory. Discrete Mathematics and Its Applications, vol. 25. CRC Press, Boca Raton, FL (2004)Google Scholar
  81. 81.
    Gupta, R.P.: The chromatic index and the degree of a graph. Notices Amer. Math. Soc. 13, Abstract 66T–429 (1966)Google Scholar
  82. 82.
    Gutman, I.: The energy of a graph. Ber. Math. Stat. Sekt. Forschungszent. Graz. 103, 1–22 (1978)Google Scholar
  83. 83.
    Gutman, I.: The energy of a graph: Old and new results. In: Betten, A., Kohnert, A., Laue, R., Wassermann, A. (eds.) Algebraic Combinatorics and Applications, pp. 196–211, Springer, Berlin (2001)Google Scholar
  84. 84.
    Gutman, I., Polansky, O.: Mathematical Concepts in Organic Chemistry. Springer-Verlag, Berlin (1986)MATHGoogle Scholar
  85. 85.
    Gutman, I., Zhou, B.: Laplacian energy of a graph. Linear Algebra Appl. 414, 29–37 (2006)MathSciNetMATHGoogle Scholar
  86. 86.
    Gutman, I., Soldatović, T., Vidović, D.: The energy of a graph and its size dependence. A Monte Carlo approach. Chem. Phys. Lett. 297, 428–432 (1998)Google Scholar
  87. 87.
    Gutman, I., Firoozabadi, S.Z., de la Pẽna, J.A., Rada, J.: On the energy of regular graphs. MATCH Commun. Math. Comput. Chem. 57, 435–442 (2007)MathSciNetMATHGoogle Scholar
  88. 88.
    Gyárfás, A., Lehel, J., Nešetril, J., Rödl, V., Schelp, R.H., Tuza, Z.: Local k-colorings of graphs and hypergraphs. J. Combin. Theory Ser. B 43, 127–139 (1987)MathSciNetMATHGoogle Scholar
  89. 89.
    Hajnal, A., Surányi, J.: Über die Auflösung von Graphen in Vollständige Teilgraphen. Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 1, 113–121 (1958)Google Scholar
  90. 90.
    Hall, P.: On representatives of subsets. J. London Math. Soc. 10, 26–30 (1935)Google Scholar
  91. 91.
    Hall, M. Jr.: Combinatorial Theory. Blaisdell, Waltham, MA (1967)MATHGoogle Scholar
  92. 92.
    Harary, F.: The determinant of the adjacency matrix of a graph. SIAM Rev 4, 202–210 (1962)MathSciNetMATHGoogle Scholar
  93. 93.
    Harary, F.: Graph Theory. Addison-Wesley, Reading, MA (1969)Google Scholar
  94. 94.
    Harary, F., Nash-Williams, C.St.J.A.: On Eulerian and Hamiltonian graphs and line graphs. Canad. Math. Bull. 8, 701–710 (1965)Google Scholar
  95. 95.
    Harary, F., Palmer, E.M.: Graphical Enumeration, Academic Press, New York (1973)MATHGoogle Scholar
  96. 96.
    Harary, F., Tutte, W.T.: A dual form of Kuratowski’s theorem. Canad. Math. Bull. 8, 17–20 (1965)MathSciNetGoogle Scholar
  97. 97.
    Harary, F., Norman, R.Z., Cartwright, D.: Structural Models: An Introduction to the Theory of Directed Graphs. Wiley, New York (1965)MATHGoogle Scholar
  98. 98.
    Hartnell, B., Rall, D.F.: On Vizing’s conjecture. Congr. Numer. 82, 87–96 (1991)MathSciNetGoogle Scholar
  99. 99.
    Hartnell, B., Rall, D.F.: Domination in Cartesian products: Vizing’s conjecture. In: Domination in Graphs, Advanced Topics, vol. 209. Monographs and Textbooks in Pure and Applied Mathematics, pp. 163–189. Marcel Dekker, New York (1998)Google Scholar
  100. 100.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998)MATHGoogle Scholar
  101. 101.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: Advanced Topics. Marcel Dekker, New York (1998)MATHGoogle Scholar
  102. 102.
    Hayward, R.B.: Weakly triangulated graphs. J. Combin. Theory Ser. B 39, 200–208 (1985)MathSciNetMATHGoogle Scholar
  103. 103.
    Heawood, P.J.: Map colour theorems. Quart. J. Math. 24, 332–338 (1890)Google Scholar
  104. 104.
    Hedetniemi, S.T.: Homomorphisms of graphs and automata. Tech. Report 03105-44-T, University of Michigan (1966)Google Scholar
  105. 105.
    Hell, P., Nešetril, J.: Graphs and homorphisms. Oxford Lecture Series in Mathematics and its Applications, vol. 28. Oxford University Press, Oxford (2004)Google Scholar
  106. 106.
    Holton, D.A., Sheehan, J.: The Petersen graph. Australian Mathematical Society Lecture Series, vol. 7, Cambridge University Press, Cambridge (1993)Google Scholar
  107. 107.
    Hougardy, S., Le, V.B., Wagler, A.: Wing-triangulated graphs are perfect. J. Graph Theory 24, 25–31 (1997)MathSciNetMATHGoogle Scholar
  108. 108.
    Ilić, A.: The energy of unitary Cayley graphs. Linear Algebra Appl. 431, 1881–1889 (2009)MathSciNetMATHGoogle Scholar
  109. 109.
    Ilić, A.: Distance spectra and distance energy of integral circulant graphs. Linear Algebra Appl. 433, 1005–1014 (2010)MathSciNetMATHGoogle Scholar
  110. 110.
    Indulal, G., Gutman, I., Vijayakumar, A.: On distance energy of graphs. MATCH Commun. Math. Comput. Chem. 60, 461–472 (2008)MathSciNetMATHGoogle Scholar
  111. 111.
    Irving, R.W., Manlove, D.F.: The b-chromatic number of a graph. Discrete Appl. Math. 91, 127–141 (1999)MathSciNetMATHGoogle Scholar
  112. 112.
    Jacobson, M.S., Kinch, L.F.: On the domination number of product graphs: I. Ars. Combin. 18, 33–44 (1984)MathSciNetMATHGoogle Scholar
  113. 113.
    Jaeger, F.: A note on sub-Eulerian graphs. J. Graph Theory 3, 91–93 (1979)MathSciNetMATHGoogle Scholar
  114. 114.
    Jaeger, F.: Nowhere-zero flow problems. In: Beineke, L.W., Wilson, R.J. (eds.) Selected Topics in Graph Theory III, pp. 71–95. Academic Press, London (1988)Google Scholar
  115. 115.
    Jaeger, F., Payan, C.: Relations du type Nordhaus–Gaddum pour le nombre d’absorption d’un graphe simple. C. R. Acad. Sci. Paris A 274, 728–730 (1972)MathSciNetMATHGoogle Scholar
  116. 116.
    Jensen, T.R., Toft, B.: Graph coloring problems. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1995)MATHGoogle Scholar
  117. 117.
    Jordan, C.: Sur les assemblages de lignes. J. Reine Agnew. Math. 70, 185–190 (1869)MATHGoogle Scholar
  118. 118.
    Jüng, H.: Zu einem Isomorphiesatz von Whitney für Graphen. Math. Ann. 164, 270–271 (1966)MathSciNetGoogle Scholar
  119. 119.
    Jünger, M., Pulleyblank, W.R., Reinelt, G.: On partitioning the edges of graphs into connected subgraphs. J. Graph Theory 9, 539–549 (1985)MathSciNetMATHGoogle Scholar
  120. 120.
    Kainen, P.C., Saaty, T.L.: The Four-Color Problem (Assaults and Conquest). Dover Publications, New York (1977)MATHGoogle Scholar
  121. 121.
    Kempe, A.: On the geographical problem of four colours. Amer. J. Math. 2, 193–200 (1879)MathSciNetGoogle Scholar
  122. 122.
    Kilpatrick, P.A.: Tutte’s first colour-cycle conjecture. Ph.D. thesis, Cape Town, (1975)Google Scholar
  123. 123.
    Klotz, W., Sander, T.: Some properties of unitary Cayley graphs. Electron. J. Combinator. 14, 1–12 (2007)MathSciNetGoogle Scholar
  124. 124.
    Koolen, J.H., Moulton, V.: Maximal energy graphs. Adv. Appl. Math. 26, 47–52 (2001)MathSciNetMATHGoogle Scholar
  125. 125.
    Kouider, M., Mahéo, M.: Some bounds for the b-chromatic number of a graph. Discrete Math. 256, 267–277 (2002)MathSciNetMATHGoogle Scholar
  126. 126.
    Kratochvíl, J., Tuza, Z., Voigt, M.: On the b-chromatic number of graphs. Lecture Notes Comput. Sci. 2573, 310–320 (2002)Google Scholar
  127. 127.
    Kruskal, J.B. Jr.: On the shortest spanning subtree of a graph and the travelling salesman problem. Proc. Amer. Math. Soc. 7, 48–50 (1956)MathSciNetMATHGoogle Scholar
  128. 128.
    Kundu, S.: Bounds on the number of disjoint spanning trees. J. Combin. Theory Ser. B 17, 199–203 (1974)MathSciNetMATHGoogle Scholar
  129. 129.
    Kuratowski, C.: Sur le problème des courbes gauches en topologie. Fund. Math. 15, 271–283 (1930)MATHGoogle Scholar
  130. 130.
    Laskar, R., Shier, D.: On powers and centers of chordal graphs. Discrete Applied Math. 6, 139–147 (1983)MathSciNetMATHGoogle Scholar
  131. 131.
    Lesniak, L.M.: Neighborhood unions and graphical properties. In: Alavi, Y., Chartrand, G., Ollermann, O.R., Schwenk, A.J. (eds.) Proceedings of the Sixth Quadrennial International Conference on the Theory and Applications of Graphs: Graph Theory, Combinatorics and Applications. Western Michigan University, pp. 783–800. Wiley, New York (1991)Google Scholar
  132. 132.
    Li, W.C.W.: Number theory with applications. Series of University Mathematics, vol. 7. World Scientific, Singapore (1996)Google Scholar
  133. 133.
    Li, X., Li, Y., Shi, Y.: Note on the energy of regular graphs. Linear Algebra Appl. 432, 1144–1146 (2010)MathSciNetMATHGoogle Scholar
  134. 134.
    Lovász, L.: Normal hypergraphs and the perfect graph conjecture. Discrete Math. 2, 253–267 (1972)MathSciNetMATHGoogle Scholar
  135. 135.
    Lovász, L.: Three short proofs in graph theory. J. Combin Theory Ser. B 19, 111–113 (1975)Google Scholar
  136. 136.
    Lovász, L., Plummer, M.D.: Matching theory. Annals of Discrete Mathematics, vol. 29. North-Holland Mathematical Studies, vol. 121 (1986)Google Scholar
  137. 137.
    Lubotzky, A., Phillips, R., Sarnak, P.: Ramanujan graphs. Combinatorica 8, 261–277 (1988)MathSciNetMATHGoogle Scholar
  138. 138.
    McKee, T.A.: Recharacterizing Eulerian: Intimations of new duality. Discrete Math. 51, 237–242 (1984)MathSciNetMATHGoogle Scholar
  139. 139.
    Meir, A., Moon, J.W.: Relations between packing and covering numbers of a tree. Pacific J. Math. 61, 225–233 (1975)MathSciNetMATHGoogle Scholar
  140. 140.
    Menger, K.: Zur allgemeinen Kurventheorie. Fund. Math. 10, 96–115 (1927)MATHGoogle Scholar
  141. 141.
    Moon, J.W.: On subtournaments of a tournament. Canad. Math. Bull. 9, 297–301 (1966)MathSciNetMATHGoogle Scholar
  142. 142.
    Moon, J.W.: Various proofs of Cayley’s formula for counting trees. In: Harary, F. (eds.) A Seminar on Graph Theory, pp. 70–78. Holt, Rinehart and Winston, Inc., New York (1967)Google Scholar
  143. 143.
    Moon, J.W.: Topics on Tournaments. Holt, Rinehart and Winston Inc., New York (1968)Google Scholar
  144. 144.
    Mycielski, J.: Sur le coloriage des graphs. Colloq. Math. 3, 161–162 (1955)MathSciNetMATHGoogle Scholar
  145. 145.
    Nash-Williams, C.St.J.A.: Edge-disjoint spanning trees of finite graphs. J. London Math. Soc. 36, 445–450 (1961)Google Scholar
  146. 146.
    Nebesky, L.: On the line graph of the square and the square of the line graph of a connected graph, Casopis. Pset. Mat. 98, 285–287 (1973)MathSciNetMATHGoogle Scholar
  147. 147.
    Nikiforov, V.: The energy of graphs and matrices. J. Math. Anal. Appl. 326, 1472–1475 (2007)MathSciNetMATHGoogle Scholar
  148. 148.
    Nordhaus, E.A., Gaddum, J.W.: On complementary graphs. Amer. Math. Monthly 63, 175–177 (1956)MathSciNetMATHGoogle Scholar
  149. 149.
    Oberly, D.J., Sumner, D.P.: Every connected, locally connected nontrivial graph with no induced claw is Hamiltonian. J. Graph Theory 3, 351–356 (1979)MathSciNetMATHGoogle Scholar
  150. 150.
    Ore, O.: Note on Hamilton circuits. Amer. Math. Monthly 67, 55 (1960)MathSciNetMATHGoogle Scholar
  151. 151.
    Ore, O.: Theory of graphs. Amer. Math. Soc. Transl. 38, 206–212 (1962)Google Scholar
  152. 152.
    Ore, O.: The Four-Color Problem. Academic Press, New York (1967)MATHGoogle Scholar
  153. 153.
    Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Upper Saddle River, NJ (1982)MATHGoogle Scholar
  154. 154.
    Parthasarathy, K.R., Ravindra, G.: The strong perfect-graph conjecture is true for K 1, 3-free graphs. J. Combin. Theory Ser. B 21, 212–223 (1976)MathSciNetMATHGoogle Scholar
  155. 155.
    Parthasarathy, K.R.: Basic Graph Theory. Tata McGraw-Hill Publishing Company Limited, New Delhi (1994)Google Scholar
  156. 156.
    Parthasarathy, K.R., Ravindra, G.: The validity of the strong perfect-graph conjecture for (K 4 − e)-free graphs. J. Combin. Theory Ser. B 26, 98–100 (1979)MathSciNetMATHGoogle Scholar
  157. 157.
    Peña, I., Rada, J.: Energy of digraphs. Linear and Multilinear Algebra 56(5), 565–579 (2008)MathSciNetMATHGoogle Scholar
  158. 158.
    Petersen, J.: Die Theorie der regulären Graphen. Acta Math. 15, 193–220 (1891)MathSciNetMATHGoogle Scholar
  159. 159.
    Prim, R.C.: Shortest connection networks and some generalizations. Bell System Techn. J. 36, 1389–1401 (1957)Google Scholar
  160. 160.
    Rall, D.F.: Total domination in categorical products of graphs. Discussiones Mathematicae Graph Theory 25, 35–44 (2005)MathSciNetMATHGoogle Scholar
  161. 161.
    Ramaswamy, H.N., Veena, C.R.: On the energy of unitary Cayley graphs. Electron. J. Combinator. 16 (2009)Google Scholar
  162. 162.
    Ram Murty, M.: Ramanujan graphs. J. Ramanujan Math. Soc. 18(1), 1–20 (2003)Google Scholar
  163. 163.
    Ram Murty, M.: Ramanujan graphs and zeta functions. Jeffery-Williams Prize Lecture. Canadian Mathematical Society, Canada (2003)Google Scholar
  164. 164.
    Ray-Chaudhuri, D.K., Wilson, R.J.: Solution of Kirkman’s schoolgirl problem. In: Proceedings of the Symposium on Mathematics, vol. 19, pp. 187–203. American Mathematical Society, Providence, RI (1971)Google Scholar
  165. 165.
    Rédei, L.: Ein kombinatorischer satz. Acta. Litt. Sci. Szeged 7, 39–43 (1934)Google Scholar
  166. 166.
    Roberts, F.S.: Graph theory and its applications to problems in society. CBMS-NSF Regional Conference Series in Mathematics. SIAM, Philadelphia (1978)Google Scholar
  167. 167.
    Sachs, H.: Über teiler, faktoren und charakteristische polynome von graphen II. Wiss. Z. Techn. Hochsch. Ilmenau 13, 405–412 (1967)MathSciNetMATHGoogle Scholar
  168. 168.
    Sampathkumar, E.: A characterization of trees. J. Karnatak Univ. Sci. 32, 192–193 (1987)MathSciNetGoogle Scholar
  169. 169.
    Schwenk, A.J., Lossers, O.P.: Solutions of advanced problems. Am. Math. Mon. 94, 885–887 (1987)MathSciNetGoogle Scholar
  170. 170.
    Serre, J.-P.: Trees. Springer-Verlag, New York (1980)MATHGoogle Scholar
  171. 171.
    Shader, B., So, W.: Skew spectra of oriented graphs. Electron. J. Combinator. 16, 1–6 (2009)MathSciNetGoogle Scholar
  172. 172.
    Shrikhande, S.S., Bhagwandas: Duals of incomplete block designs. J. Indian Stat. Assoc. 3, 30–37 (1965)MathSciNetGoogle Scholar
  173. 173.
    Stevanović, D., Stanković, I.: Remarks on hyperenergetic circulant graphs. Linear Algebra Appl. 400, 345–348 (2005)MATHGoogle Scholar
  174. 174.
    Sumner, D.P.: Graphs with 1-factors. Proc. Amer. Math. Soc. 42, 8–12 (1974)MathSciNetMATHGoogle Scholar
  175. 175.
    Toida, S.: Properties of an Euler graph. J. Franklin. Inst. 295, 343–346 (1973)MathSciNetMATHGoogle Scholar
  176. 176.
    Trinajstic, N.: Chemical Graph Theory—Volume I. CRC Press, Boca Raton, FL (1983)Google Scholar
  177. 177.
    Trinajstic, N.: Chemical Graph Theory—Volume II. CRC Press, Boca Raton, FL (1983)Google Scholar
  178. 178.
    Tucker, A.: The validity of perfect graph conjecture for K 4-free graphs. In: Berge, C., Chvátal, V. (eds.) Topics on Perfect Graphs, vol. 21, pp. 149–157 (1984)Google Scholar
  179. 179.
    Tutte, W.T.: The factorization of linear graphs. J. London Math. Soc. 22, 107–111 (1947)MathSciNetMATHGoogle Scholar
  180. 180.
    Tutte, W.T.: A theorem on planar graphs. Trans. Amer. Math. Soc. 82, 570–590 (1956)MathSciNetGoogle Scholar
  181. 181.
    Tutte, W.T.: On the problem of decomposing a graph into n connected factors. J. London Math. Soc. 36, 221–230 (1961)MathSciNetMATHGoogle Scholar
  182. 182.
    Vizing, V.G.: The Cartesian product of graphs. Vycisl/Sistemy 9, 30–43 (1963)MathSciNetGoogle Scholar
  183. 183.
    Vizing, V.G.: On an estimate of the chromatic class of a p-graph (in Russian). Diskret. Analiz. 3, 25–30 (1964)MathSciNetGoogle Scholar
  184. 184.
    Vizing, V.G.: A bound on the external stability number of a graph. Dokl. Akad. Nauk. SSSR 164, 729–731 (1965)MathSciNetGoogle Scholar
  185. 185.
    Vizing, V.G.: Some unsolved problems in graph theory. Uspekhhi Mat. Nauk. 23(6), 117–134 (1968)MathSciNetMATHGoogle Scholar
  186. 186.
    Wagner, K.: Über eine eigenschaft der ebenen komplexe. Math. Ann. 114, 570–590 (1937)MathSciNetGoogle Scholar
  187. 187.
    Walikar, H.B., Acharya, B.D., Sampathkumar, E.: Recent developments in the theory of domination in graphs. MRI Lecture Notes in Mathematics, vol. 1. Mehta Research Institue, Allahabad (1979)Google Scholar
  188. 188.
    Walikar, H.B., Ramane, H.S., Hampiholi, P.R.: On the energy of a graph. In: Mulder, H.M., Vijayakumar, A., Balakrishnan, R. (eds.) Graph Connections, pp. 120–123. Allied Publishers, New Delhi (1999)Google Scholar
  189. 189.
    Walikar, H.B., Gutman, I., Hampiholi, P.R., Ramane, H.S.: Non-hyperenergetic graphs. Graph Theory Notes New York 41, 14–16 (2001)MathSciNetGoogle Scholar
  190. 190.
    Walikar, H.B., Ramane, H.S., Jog, S.R.: On an open problem of R. Balakrishnan and the energy of products of graphs. Graph Theory Notes New York 55, 41–44 (2008)MathSciNetGoogle Scholar
  191. 191.
    Welsh, D.J.A.: Matroid Theory. Academic Press, London (1976)MATHGoogle Scholar
  192. 192.
    West, D.B.: Introduction to Graph Theory, 2nd ed. Prentice Hall, New Jersey (2001)Google Scholar
  193. 193.
    Whitney, H.: Congruent graphs and the connectivity of graphs. Amer. J. Math. 54, 150–168 (1932)MathSciNetGoogle Scholar
  194. 194.
    Yap, H.P.: Some topics in graph theory. London Mathematical Society Lecture Notes Series, vol. 108, Cambridge University Press, Cambridge (1986)Google Scholar
  195. 195.
    Zykov, A.A.: On some properties of linear complexes (in Russian). Math. Sbornik N. S. 24, 163–188 (1949); Amer. Math Soc. Trans. 79 (1952)Google Scholar

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© Springer Science+Business Media New York 2012

Authors and Affiliations

  • R. Balakrishnan
    • 1
  • K. Ranganathan
  1. 1.Department of MathematicsBharathidasan UniversityTiruchirappalliIndia

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