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.