In this short introductory course to graph theory, possibly one of the most propulsive areas of contemporary mathematics, some of the basic graph-theoretic concepts together with some open problems in this scientific field are presented.
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References
Alspach B (1979) Hamiltonian cycles in vertex-transitive graphs of order 2p. In: Proceedings of the tenth southeastern conference on combinatorics, graph theory and computing, Florida Atlantic University, Boca Raton, FL, 1979, pp 131–139. Congress Numer XXIII—XX, Utilitas Math, Winnipeg, MB
Alspach B (1983) The classification of hamiltonian generalized Petersen graphs. J Combin Theory B 4:293–312
Alspach B (1989a) Hamilton cycles in metacirculant graphs with prime power cardinal blocks. In: Graph theory in memory of G. A. Dirac (Sandbjerg, 1985). North-Holland, Amsterdam, pp 7–16. Ann Discrete Math 41
Alspach B (1989b) Lifting Hamilton cycles of quotient graphs. Discrete Math 78:25–36
Alspach B, Chen CC, McAvaney K (1996) On a class of Hamiltonian laceable 3-regular graphs. Discrete Math 151:19–38
Alspach B, Durnberger E, Parsons TD (1985) Hamilton cycles in metacirculant graphs with prime cardinality blocks. In: Cycles in graphs (Burnaby, B.C., 1982). North-Holland, Amsterdam, pp 27–34. Ann Discrete Math 27
Alspach B, Locke S, Witte D (1990) The Hamilton spaces of Cayley graphs on abelian groups. Discrete Math 82:113–126
Alspach B, Parsons TD (1982) On Hamiltonian cycles in metacirculant graphs, Algebraic and geometric combinatorics. Ann Discrete Math 15:1–7
Alspach B, Qin YS (2001) Hamilton-connected Cayley graphs on Hamiltonian groups. European J Combin 22:777–787
Alspach B, Zhang CQ (1989) Hamilton cycles in cubic Cayley graphs on dihedral groups. Ars Combin 28:101–108
Betten A, Brinkmann G, Pisanski T (2000) Counting symmetric configurations v3. Discrete Appl Math 99:331–338
Biggs N (1993) Algebraic graph theory, 2nd edn. Cambridge University Press, Cambridge
Bouwer IZ (1968) An edge but not vertex transitive cubic graph. Bull Canad Math Soc 11:533–535
Bouwer IZ (1970) Vertex and edge-transitive but not 1-transitive graphs. Canad Math Bull 13:231–237
Bouwer IZ (ed) (1988) The Foster census. Charles Babbage Research Centre, Winnipeg
Brinkmann G, Steffen E (1998) Snarks and reducibility. Ars Combin 50:292–296
Brooks RL (1941) On coloring the nodes of a network. Proc Cambridge Philos Soc 37:194–197
Cameron PJ (ed) (1997) Problems from the fifteenth British combinatorial conference. Discrete Math 167/168:605–615
Chen YQ (1998) On Hamiltonicity of vertex-transitive graphs and digraphs of order p4. J Combin Theory B 72:110–121
Conder MDE (2006) Trivalent symmetric graphs on up to 2048 vertices. http://www. math.auckland.ac.nz/conder/symmcubic2048list.txt
Conder MDE, Dobcsanyi P (2002) Trivalent symmetric graphs on up to 768 vertices. J Com-bin Math Combin Comp 40:41–63
Conder MDE, Lorimer PJ (1989) Automorphism groups of symmetric graphs of valency 3. J Combin Theory B 47:60–72
Conder MDE, Malnič A, Marušič D, Pisanski T, Potočnik P (2005) The edge-transitive but not vertex-transitive cubic graph on 112 vertices. J Graph Theory 50:25–42
Conder MDE, Malnič A, Marušič D, Potočnik P (2006) A census of semisymmetric cubic graphs on up to 768 vertices. J Algebraic Combin 23:255–294
Conder MDE, Marušič D (2003) A tetravalent half-arc-transitive graph with nonabelian vertex stabilizer. J Combin Theory B 88:67–76
Conder MDE, Morton M (1995) Classification of trivalent symmetric graphs of small order. Australas J Combin 11:139–149
Conder MDE, Nedela R (2007) Symmetric cubic graphs of small girth. J Combin Theory B 97:757–768
Conder MDE, Nedela R (2008) A more detailed classification of symmetric cubic graphs (preprint)
Coxeter HSM (1950) Self-dual configurations and regular graphs. Bull Am Math Soc 56:413–455
Curran S, Gallian JA (1996) Hamiltonian cycles and paths in Cayley graphs and digraphs — a survey. Discrete Math 156:1–18
Dirac GA (1952) Some theorems on abstract graphs. Proc London Math Soc (3) 2:69–81
Dobson E, Gavlas H, Morris J, Witte D (1998) Automorphism groups with cyclic commutator subgroup and Hamilton cycles. Discrete Math 189:69–78
Dobson E, Malnič A, Marušič D (2007) Semiregular automorphisms of vertex-transitive graphs of certain valencies. J Combin Theory B 97:371–380
Du SF, Xu MY (1998) Vertex-primitive 1/2-arc-transitive graphs of smallest order. Comm Algebra 27:163–171
Du SF, Xu MY (2000) A classification of semisymmetric graphs of order 2pq. Comm Algebra 28:2685–2715
Folkman J (1967) Regular line-symmetric graphs. J Combin Theory 3:215–232
Glover HH, Marušič D (2007) Hamiltonicity of cubic Cayley graph. J Eur Math Soc 9:775–787
Godsil C, Royle G (2001) Algebraic graph theory. Springer, New York
Gropp H (1993) Configurations and graphs. Discrete Math 111:269–276
Harary F (1969) Graph theory. Addison, Reading, MA
Kochol M (1996) A cyclically 6-edge-connected snark of order 118. Discrete Math 161:297–300
Landau HG (1953) On dominance relations and the structure of animal societies: III The condition for a score structure. Bull Math Biol 15:143–148
Li CH, Sims HS (2001) On half-transitive metacirculant graphs of prime-power order. J Com-bin Theory B 81:45–57
Kutnar K, Marušič D (2008) Hamiltonicity of vertex-transitive graphs of order 4p. Eur J Combin 29(2):423–438. doi:10.1016/j.ejc.2007.02.002
Kutnar K, Šparl P (2008) Hamilton cycles and paths in vertex-transitive graphs of order 6p. Discrete Math (submitted)
Lovász L (1970) Combinatorial structures and their applications. In: Proc. Calgary Internat. Conf., Calgary, Alberta, 1969, pp 243Š246. Problem 11. Gordon and Breach, New York
Máčajová E, Škoviera M (2006) Irreducible snarks of given order and cyclic connectivity. Discrete Math 306:779–791
Malnič A, Marušič D (1999) Constructing 4-valent 1/2-transitive graphs with a nonsolvable group. J Combin Theory B 75:46–55
Malnič A, Marušič D (2002) Constructing 1/2-transitive graphs of valency 4 and vertex stabilizer ℤ2 ×ℤ2. Discrete Math 245:203–216
Malnič A, Marušič D, Wang C (2004) Cubic edge-transitive graphs of order 2p3. Discrete Math 274:187–198
Marušič D (1981) On vertex symmetric digraphs. Discrete Math 36:69–81
Marušič D (1983) Hamilonian circuits in Cayley graphs. Discrete Math 46:49–54
Marušič D (1985) Vertex transitive graphs and digraphs of order p k. In: Cycles in graphs (Burnaby BC, 1982). North-Holland, Amsterdam, pp 115–128. Ann Discrete Math 27
Marušič D (1987) Hamiltonian cycles in vertex symmetric graphs of order 2p2. Discrete Math 66:169–174
Marušič D (1988) On vertex-transitive graphs of order qp. J Combin Math Combin Comput 4:97–114
Marušič D (1998) Half-transitive group actions on finite graphs of valency 4. J Combin Theory B 73:41–76
Marušič D (2000) Constructing cubic edge- but not vertex-transitive graphs. J Graph Theory 35:152–160
Marušič D (2005) Quartic half-arc-transitive graphs with large vertex stabilizers. Discrete Math 299:180–193
Marušič D, Parsons TD (1982) Hamiltonian paths in vertex-symmetric graphs of order 5p. Discrete Math 42:227–242
Marušič D, Scapellato R (1998) Permutation groups, vertex-transitive digraphs and semireg-ular automorphisms. Eur J Combin 19:707–712
Nedela R, Škoviera M (1996) Decompositions and Reductions of Snarks. J Graph Theory 22:253–279
Ore O (1960) Note on Hamilton circuits. Am Math Monthly 67:55
Parker CW (2007) Semisymmetric cubic graphs of twice odd order. Eur J Combin 28:572–591
Passman D (1968) Permutation groups. Benjamin, New York
Sabidussi G (1958) On a class of fixed-point-free graphs. Proc Am Math Soc 9:800–804
Steffen E (1998) Classifications and characterizations of snarks. Discrete Math 188:183–203
Šparl P (2008) A classification of tightly attached half-arc-transitive graphs of valency 4. J Combin Theory B, doi:10.1016/j.jctb.2008.01.001, in press
Turner J (1967) Point-symmetric graphs with a prime number of points. J Combin Theory 3:136–145
Tutte WT (1947) A family of cubical graphs. Proc Cambridge Philos Soc 43:459–474
Tutte WT (1966) Connectivity in graphs. University of Toronto Press, Toronto
Vizing VG (1964) On an estimate of the chromatic class of a p-graph. Metody Diskret Analiz 3:25–30
Wielandt H (1964) Finite permutation groups. Academic, New York
Xu J (2008) Semiregular automorphisms of arc-transitive graphs with valency pq. Eur J Com-bin 29(3):622–629. doi:10.1016/j.ejc.2007.04.008
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Kutnar, K., Marušič, D. (2009). Some Topics in Graph Theory. In: Naimzada, A.K., Stefani, S., Torriero, A. (eds) Networks, Topology and Dynamics. Lecture Notes in Economics and Mathematical Systems, vol 613. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68409-1_1
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