Chvátal first learned of graph theory in 1964, on finding a book by Claude Berge in a Pilsen bookstore [8] and much of his research involves graph theory:
His first mathematical publication, at the age of 19, concerned directed graphs that cannot be mapped to themselves by any nontrivial graph homomorphism[9]
A 1972 paper [11] relating Hamiltonian cycles to connectivity and maximum independent set size of a graph, earned Chvátal his Erdős number of 1. Specifically, if there exists an s such that a given graph is s-vertex-connected and has no (s + 1)-vertex independent set, the graph must be Hamiltonian. Avis et al.[4] tell the story of Chvátal and Erdős working out this result over the course of a long road trip, and later thanking Louise Guy "for her steady driving."
In a 1973 paper,[12] Chvátal introduced the concept of graph toughness, a measure of graph connectivity that is closely connected to the existence of Hamiltonian cycles. A graph is t-tough if, for every k greater than 1, the removal of fewer than tk vertices leaves fewer than k connected components in the remaining subgraph. For instance, in a graph with a Hamiltonian cycle, the removal of any nonempty set of vertices partitions the cycle into at most as many pieces as the number of removed vertices, so Hamiltonian graphs are 1-tough. Chvátal conjectured that 3/2-tough graphs, and later that 2-tough graphs, are always Hamiltonian; despite later researchers finding counterexamples to these conjectures, it still remains open whether some constant bound on the graph toughness is enough to guarantee Hamiltonicity.[13]
Some of Chvátal's work concerns families of sets, or equivalently hypergraphs, a subject already occurring in his Ph.D. thesis, where he also studied Ramsey theory.
In a 1972 conjecture that Erdős called "surprising" and "beautiful",[14] and that remains open (with a $10 prize offered by Chvátal for its solution) [15][16] he suggested that, in any family of sets closed under the operation of taking subsets, the largest pairwise-intersecting subfamily may always be found by choosing an element of one of the sets and keeping all sets containing that element.
Chvátal first became interested in linear programming through the influence of Jack Edmonds while Chvátal was a student at Waterloo.[4] He quickly recognized the importance of cutting planes for attacking combinatorial optimization problems such as computing maximum independent sets and, in particular, introduced the notion of a cutting-plane proof.[18][19][20][21] At Stanford in the 1970s, he began writing his popular textbook, Linear Programming, which was published in 1983.[4]
Cutting planes lie at the heart of the branch and cut method used by efficient solvers for the traveling salesman problem. Between 1988 and 2005, the team of David L. Applegate, Robert E. Bixby, Vašek Chvátal, and William J. Cook developed one such solver, Concorde.[22][23] The team was awarded The Beale-Orchard-Hays Prize for Excellence in Computational Mathematical Programming in 2000 for their ten-page paper [24] enumerating some of Concorde's refinements of the branch and cut method that led to the solution of a 13,509-city instance and it was awarded the Frederick W. Lanchester Prize in 2007 for their book, The Traveling Salesman Problem: A Computational Study.
Vašek Chvátal (1983). Linear Programming. W.H. Freeman. ISBN 978-0-7167-1587-0.. Japanese translation published by Keigaku Shuppan, Tokyo, 1986.
C. Berge and V. Chvátal (eds.) (1984). Topics on Perfect Graphs. Elsevier. ISBN 978-0-444-86587-8. {{cite book}}: |author= has generic name (help)
David L. Applegate; Robert E. Bixby; Vašek Chvátal; William J. Cook (2007). The Traveling Salesman Problem: A Computational Study. Princeton University Press. ISBN 978-0-691-12993-8.[33]
Vašek Chvátal, ed. (2011). Combinatorial Optimization: Methods and Applications. IOS Press. ISBN 978-1-60750-717-8.
Vašek Chvátal (2021). Discrete Mathematical Charms of Paul Erdős. A Simple Introduction. Cambridge University Press. ISBN 978-1-108-92740-6.
^V. Chvátal; David A. Klarner; D.E. Knuth (1972), "Selected combinatorial research problems" (PDF), Computer Science Department, Stanford University, Stan-CS-TR-72-292: Problem 25
^Chvátal, Vašek, A conjecture in extremal combinatorics
^"A greedy heuristic for the set-covering problem", Mathematics of Operations Research, 1979
^Chvátal, Václav (1973), "Edmonds polytopes and weakly hamiltonian graphs", Mathematical Programming, 5: 29–40, doi:10.1007/BF01580109, S2CID 8140217,
^Chvátal, Václav (1973), "Edmonds polytopes and a hierarchy of combinatorial problems", Discrete Mathematics, 4 (4): 305–337, doi:10.1016/0012-365x(73)90167-2,
^Chvátal, Václav (1975), "Some linear programming aspects of combinatorics" (PDF), Congressus Numerantium, 13: 2–30,
^Chvátal, V. (1975), "On certain polytopes associated with graphs", Journal of Combinatorial Theory, Series B, 18 (2): 138–154, doi:10.1016/0095-8956(75)90041-6.
^Math Problem, Long Baffling, Slowly Yields. New York Times, Mar. 12, 1991.
^Applegate, David; Bixby, Robert; Chvátal, Vašek; Cook, William (1998), "On the Solution of Traveling Salesman Problems", Documenta Mathematica, Extra Volume ICM III
^Weisstein, Eric W. "Art Gallery Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ArtGalleryTheorem.html
^Diagonals: Part I 4. Art gallery problems, AMS Feature Column by Joseph Malkevitch
^Chvátal, Václav; Sankoff, David (1975), "Longest common subsequences of two random sequences", Journal of Applied Probability, 12 (2): 306–315, doi:10.2307/3212444, JSTOR 3212444, S2CID 250345191.
^Chvátal, Vašek; Szemerédi, Endre (1988), "Many hard examples for resolution", Journal of the ACM, 35 (4): 759–768, doi:10.1145/48014.48016, S2CID 2526816.
^Borchers, Brian (March 25, 2007). "Review of The Traveling Salesman Problem: A Computational Study". MAA Reviews, Mathematical Association of America.
