Best Monotone Degree Conditions for Graph Properties: A Survey

We survey sufficient degree conditions, for a variety of graph properties, that are best possible in the same sense that Chvátal’s well-known degree condition for hamiltonicity is best possible.

[1]  Michael J. Mossinghoff,et al.  Combinatorics and graph theory , 2000 .

[2]  G. Dirac.  Some Theorems on Abstract Graphs , 1952 .

[3]  D. R. Lick,et al.  n-Hamiltonian graphs , 1970 .

[4]  D. West.  Introduction to Graph Theory , 1995 .

[5]  Vasek Chvátal,et al.  Tough graphs and hamiltonian circuits , 1973, Discret. Math..

[6]  Mehdi Behzad,et al.  Graphs and Digraphs , 1981, The Mathematical Gazette.

[7]  D. R. Woodall.  k-Factors and Neighbourhoods of Independent Sets in Graphs , 1990 .

[8]  Edward F. Schmeichel,et al.  A simple proof of a theorem of Jung , 1990, Discret. Math..

[9]  Wayne Goddard,et al.  The binding number of a graph and its cliques , 2009, Discret. Appl. Math..

[10]  S. Louis Hakimi,et al.  Recognizing tough graphs is NP-hard , 1990, Discret. Appl. Math..

[11]  C. Nash-Williams,et al.  Hamiltonian arcs and circuits , 1971 .

[12]  Douglas R. Woodall,et al.  Toughness and binding number , 2014, Discret. Appl. Math..

[13]  S. Louis Hakimi,et al.  Sufficient degree conditions for k‐edge‐connectedness of a graph , 2009, Networks.

[14]  Degree sequences and edge connectivity , 2017 .

[15]  John Adrian Bondy,et al.  A method in graph theory , 1976, Discret. Math..

[16]  Edward F. Schmeichel,et al.  Degree Sequences and the Existence of k-Factors , 2012, Graphs Comb..

[17]  F. T Boesch.  The strongest monotone degree condition for n-connectedness of a graph , 1974 .

[18]  I. Anderson.  Perfect matchings of a graph , 1971 .

[19]  G. Hardy,et al.  Asymptotic Formulaæ in Combinatory Analysis , 1918 .

[20]  D. R. Woodall.  The binding number of a graph and its Anderson number , 1973 .

[21]  Binding number, cycles and complete graphs , 1981 .

[22]  Chính T. Hoàng,et al.  Hamiltonian degree conditions for tough graphs , 1995, Discret. Math..

[23]  S. L. Hakimi,et al.  A Note on the Vertex Arboricity of a Graph , 1989, SIAM J. Discret. Math..

[24]  G. Szekeres,et al.  An inequality for the chromatic number of a graph , 1968 .

[25]  Ciping Chen.  Binding number and toughness for matching extension , 1995, Discret. Math..

[26]  Hudson V. Kronk,et al.  A note on K-path hamiltonian graphs , 1969 .

[27]  Edward F. Schmeichel,et al.  Hamiltonian degree conditions which imply a graph is pancyclic , 1990, J. Comb. Theory, Ser. B.

[28]  William H. Cunningham.  Computing the binding number of a graph , 1990, Discret. Appl. Math..

[29]  Gary Chartrand,et al.  A sufficient condition for n -connectedness of graphs , 1968 .

[30]  Claude Berge,et al.  Graphs and Hypergraphs , 2021, Clustering.

[31]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[32]  G. Chartrand,et al.  The Point‐Arboricity of Planar Graphs , 1969 .

[33]  H. A. Jung.  On Maximal Circuits in Finite Graphs , 1978 .

[34]  Edward F. Schmeichel,et al.  Binding Number, Minimum Degree, and Cycle Structure in Graphs , 2012, J. Graph Theory.

[35]  V. Chvátal.  On Hamilton's ideals , 1972 .

[36]  H. Wilf.  The Eigenvalues of a Graph and Its Chromatic Number , 1967 .

[37]  Douglas R. Woodall,et al.  Triangles and Neighbourhoods of Independent Sets in Graphs , 2000, J. Comb. Theory, Ser. B.

[38]  R. Shi.  The binding number of a graph and its pancyclism , 1987 .

[39]  Douglas R. Woodall.  A sufficient condition for hamiltonian circuits , 1978, J. Comb. Theory, Ser. B.

[40]  R. L. Brooks.  On Colouring the Nodes of a Network , 1941 .

[41]  Sufficient degree conditions for k-edge-connectedness of a graph , 2009 .

[42]  Edward F. Schmeichel,et al.  Toughness and Vertex Degrees , 2009, J. Graph Theory.

[43]  Owen Murphy,et al.  Lower bounds on the stability number of graphs computed in terms of degrees , 1991, Discret. Math..

[44]  D. J. A. Welsh,et al.  An upper bound for the chromatic number of a graph and its application to timetabling problems , 1967, Comput. J..

[45]  D. Bauera,et al.  Not every 2-tough graph is Hamiltonian ( , 1999 .

[46]  W. T. Tutte.  A Short Proof of the Factor Theorem for Finite Graphs , 1954, Canadian Journal of Mathematics.

[47]  P. Katerinis,et al.  BINDING NUMBERS OF GRAPHS AND THE EXISTENCE OF k-FACTORS , 1987 .

[48]  Tomasz Traczyk,et al.  ON n-HAMILTONIAN GRAPHS OF MINIMAL SIZE , 1981 .

[49]  Edward F. Schmeichel,et al.  Best monotone degree conditions for binding number , 2011, Discret. Math..

[50]  Bauer,et al.  Improving Theorems in a Best Monotone Sense , 2013 .

[51]  G. Hardy,et al.  Asymptotic formulae in combinatory analysis , 1918 .

[52]  Douglas R. Woodall,et al.  Best monotone degree conditions for binding number and cycle structure , 2015, Discret. Appl. Math..