Abstract
Some recent work on the relationship between an integer sequence and various graph theoretic properties which a realization of the sequence may or must possess is surveyed.
This work is supported by the Air Force Office of Scientific Research, Air Force Systems Command, under Grant AFOSR 71-2103 (D).
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References
S. R. Alpert and J. L. Gross, Graph embedding problems, Amer Math. Monthly 82 (1975), 835–837.
C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam, 1973.
L. W. Beineke and E. F. Schmeichel, On degrees and cycles in graphs, J. Networks (to appear).
F. T. Boesch, The strongest monotone degree condition for n-connectedness of a graph, J. Combinatorial Theory Ser. B. 16 (1974), 162–165.
F. T. Boesch and F. Harary, Unicyclic realizability of a degree list, J. Networks (to appear).
J. A. Bondy, Properties of graphs with constraints on degrees, Studia Sci. Math. Hungar. 4 (1966), 473–475.
J. A. Bondy, Pancyclic graphs I, J. Combinatorial Theory 11 (1971), 80–84.
J. A. Bondy and V. Chvà tal, A method in graph theory (to appear).
R. Bowen, On the sums of valences in planar graphs, Can. Math. Bull. 9 (1966), 111–114.
G. Chartrand, S. F. Kapoor, and H. V. Kronk, A sufficient condition for n-connectedness of a graph, Mathematika 15 (1968), 51–52.
W. Chou and H. Frank, Survivable communication networks and terminal capacity matrix, IEEE Trans. Circuit Theory CT-17 (1970), 192–197.
V. Chvatal, Planarity of graphs with given degrees of vertices, Nieuw. Arch. Wisk. 17 (1969), 47–60.
V. Chvátal, On Hamilton's ideals, J. Combinatorial Theory Ser. B. 12 (1972), 163–168.
V. Chvátal, New directions in Hamiltonian graph theory, in "New Directions in the Theory of Graphs" (Proceedings of the 1971 Ann Arbor Graph Theory Conference, F. Harary, ed.) Academic Press, New York, 1973.
C. R. J. Clapham and D. J. Kleitman, The degree sequences of self-complimentary graphs, J. Combinatorial Theory Ser. B. 20 (1976), 67–74.
G. A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. (3) 2 (1952), 69–81.
J. Edmonds, Existence of k-edge connected ordinary graphs with prescribed degrees, J. of Research of the National Bureau of Standards, 68B (1964), 73–74.
P. Erdös and T. Gallai, Graphs with prescribed degrees of vertices (Hungarian), Mat. Lapok 11 (1960), 264–274.
E. Etourneau, Existence and connectivity of planar graphs having 12 vertices of degree 5 and n-12 vertices of degree 6, Infinite and Finite Sets, Colloq. Math. Soc. J. Bolyai, (A. Hajnal et al, ed.) North-Holland, 1975.
B. Grünbaum and T. Motzkin, The number of hexagons and the simplicity of geodesics on certain polyhedra, Can. J. Math. 15 (1963), 744–751.
B. Grünbaum, Convex Polytopes, Wiley-Interscience, New York, 1967.
B. Grünbaum, Problem No. 2, Combinatorial Structures and Their Applications (Proceedings of the Calgony Conference, June, 1969, Richard Guy, ed.), Gordon and Breach, New York, 1971.
S. L. Hakimi, On realizability of a set of integers as degrees of the vertices of a linear graph, I, J. SIAM 10 (1962), 496–502.
S. L. Hakimi, On the realizability of a set of integers as degrees of the vertices of a linear graph, II, J. SIAM 11 (1963), 135–147.
S. L. Hakimi, On the existence of graphs with prescribed degrees and connectivity, J. SIAM Appl. Math. 26 (1974), 154–164.
F. Harary, Graph Theory, Addison-Wesley, Reading, Mass., 1969.
V. Havel, A remark on the existence of finite graphs (Hungarian), Casopis Pest. Mat. 80 (1955), 477–480.
A. Hawkins, A. Hill, J. Reeve, and J. Tyrell, On certain polyhedra, Math. Gazette 50 (1966), 140–144.
D. J. Kleitman and D. L. Wang, Algorithms for constructing graphs and digraphs with given valences and factors, Discrete Math. 6 (1973), 79–88.
S. Kundu, The k-factor conjecture is true, Discrete Math. 6 (1973), 367–376.
D. R. Lick, A sufficient condition for Hamiltonian connectedness, J. Combinatorial Theory 8 (1970), 444–445.
L. Lovász, Valencies of graphs with 1-factors, Periodica Math. Hungar. 5(2) (1974), 149–151.
M. Meyniel, Une condition suffisante d'existence d'un circuit Hamiltonian dans un graphe oriente, J. Combinatorial Theory Ser. B 14 (1973), 137–147.
C. St. J. A. Nash-Williams, Unexplored and semi-explored territories in graph theory, in "New Directions in the Theory of Graphs" (Proceedings of the 1971 Ann Arbor Graph Theory Conference, F. Harary, ed.) Academic Press, New York, 1973.
C. St. J. A. Nash-Williams, On Hamiltonian circuits in finite graphs, Proc. Amer. Math. Soc. 17 (1966), 466–467.
O. Ore, Note on Hamilton circuits, Amer. Math. Monthly 67 (1960), 55.
A. B. Owens, On the planarity of regular incidence sequences, J. Combinatorial Theory 11 (1971), 201–212.
A. Patrinos and S. L. Hakimi, Relations between graphs and integer-pair sequences, to appear in Discrete Mathematics.
L. Posa, A theorem concerning Hamilton lines, Magyar Tud. Akas. Mat. Kutáto Int. Közl 7 (1962), 225–226.
S. B. Rao and A. R. Rao, Existence of triconnected graphs with prescribed degrees, Pacific J. Math. 33 (1970), 203–207.
A. R. Rao and S. B. Rao, On factorable degree sequences, J. Combinatorial Theory Ser. B. 13 (1972), 185–191.
G. Ringel, Selbst Komplementäre Graphen, Arch. Math. 14 (1963), 354–358.
G. Ringel, Map Color Theorem, Springer-Verlag, New York, 1974.
H. Sachs, Über Selbst Komplementäre Graphen, Publ. Math. Debrecen 9 (1962), 270–288.
E. F. Schmeichel and S. L. Hakimi, Pancyclic graphs and a conjecture of Bondy and Chvátal, J. Combinatorial Theory Ser. B. 17 (1974), 22–34.
E. F. Schmeichel and S. L. Hakimi, On planar graphical degree sequences, SIAM J. Applied Math. (to appear).
E. F. Schmeichel and S. L. Hakimi, On the existence of a traceable graph with prescribed vertex degrees (in preparation).
J. K. Senior, Partitions and their representative graphs, Amer. J. Math. 73 (1951), 663–689.
W. T. Tutte, A family of cubical graphs, Proc. of Comb. Phil. Soc. 43 (1947), 459–474.
D. L. Wang and D. J. Kleitman, On the existence of n-connected graphs with prescribed degrees (n≥2), Networks 3 (1973), 225–239.
D. L. Wang D. J. Kleitman, A note on n-edge-connectivity, unpublished note.
D. L. Wang, Construction of a maximally edge-connected graph with prescribed degrees, Stud. Appl. Math. 55(1) (1976), 87–92.
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Hakimi, S.L., Schmeichel, E.F. (1978). Graphs and their degree sequences: A survey. In: Alavi, Y., Lick, D.R. (eds) Theory and Applications of Graphs. Lecture Notes in Mathematics, vol 642. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0070380
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DOI: https://doi.org/10.1007/BFb0070380
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