Read's conjecture

Last updated

Read's conjecture is a conjecture, first made by Ronald Read, about the unimodality of the coefficients of chromatic polynomials in the context of graph theory. [1] [2] In 1974, S. G. Hoggar tightened this to the conjecture that the coefficients must be strongly log-concave. Hoggar's version of the conjecture is called the Read–Hoggar conjecture. [3] [4]

The Read–Hoggar conjecture had been unresolved for more than 40 years before June Huh proved it in 2009, during his PhD studies, using methods from algebraic geometry. [1] [5] [6] [7]

Related Research Articles

<span class="mw-page-title-main">Petersen graph</span> Cubic graph with 10 vertices and 15 edges

In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring.

In graph theory, an undirected graph H is called a minor of the graph G if H can be formed from G by deleting edges, vertices and by contracting edges.

<span class="mw-page-title-main">Meyniel graph</span> Graph where all odd cycles of length ≥ 5 has 2+ chords

In graph theory, a Meyniel graph is a graph in which every odd cycle of length five or more has at least two chords. The chords may be uncrossed or they may cross each other, as long as there are at least two of them.

The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at each (triennial) International Symposium of the MOS. Originally, the prizes were paid out of a memorial fund administered by the AMS that was established by friends of the late Delbert Ray Fulkerson to encourage mathematical excellence in the fields of research exemplified by his work. The prizes are now funded by an endowment administered by MPS.

<span class="mw-page-title-main">Paul Seymour (mathematician)</span> British mathematician

Paul D. Seymour is a British mathematician known for his work in discrete mathematics, especially graph theory. He was responsible for important progress on regular matroids and totally unimodular matrices, the four colour theorem, linkless embeddings, graph minors and structure, the perfect graph conjecture, the Hadwiger conjecture, claw-free graphs, χ-boundedness, and the Erdős–Hajnal conjecture. Many of his recent papers are available from his website.

The Journal of Combinatorial Theory, Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. Series A is concerned primarily with structures, designs, and applications of combinatorics. Series B is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as JCTA and JCTB.

In graph theory, a string graph is an intersection graph of curves in the plane; each curve is called a "string". Given a graph G, G is a string graph if and only if there exists a set of curves, or strings, such that the graph having a vertex for each curve and an edge for each intersecting pair of curves is isomorphic to G.

In graph theory, an area of mathematics, an equitable coloring is an assignment of colors to the vertices of an undirected graph, in such a way that

Zoltán Füredi is a Hungarian mathematician, working in combinatorics, mainly in discrete geometry and extremal combinatorics. He was a student of Gyula O. H. Katona. He is a corresponding member of the Hungarian Academy of Sciences (2004). He is a research professor of the Rényi Mathematical Institute of the Hungarian Academy of Sciences, and a professor at the University of Illinois Urbana-Champaign (UIUC).

<span class="mw-page-title-main">Generalized Petersen graph</span> Family of cubic graphs formed from regular and star polygons

In graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. They include the Petersen graph and generalize one of the ways of constructing the Petersen graph. The generalized Petersen graph family was introduced in 1950 by H. S. M. Coxeter and was given its name in 1969 by Mark Watkins.

<span class="mw-page-title-main">Meredith graph</span>

In the mathematical field of graph theory, the Meredith graph is a 4-regular undirected graph with 70 vertices and 140 edges discovered by Guy H. J. Meredith in 1973.

<span class="mw-page-title-main">Ellingham–Horton graph</span>

In the mathematical field of graph theory, the Ellingham–Horton graphs are two 3-regular graphs on 54 and 78 vertices: the Ellingham–Horton 54-graph and the Ellingham–Horton 78-graph. They are named after Joseph D. Horton and Mark N. Ellingham, their discoverers. These two graphs provide counterexamples to the conjecture of W. T. Tutte that every cubic 3-connected bipartite graph is Hamiltonian. The book thickness of the Ellingham-Horton 54-graph and the Ellingham-Horton 78-graph is 3 and the queue numbers 2.

<span class="mw-page-title-main">György Hajós</span> Hungarian mathematician

György Hajós was a Hungarian mathematician who worked in group theory, graph theory, and geometry.

<span class="mw-page-title-main">Medial graph</span> Edge-face adjacencies in another graph

In the mathematical discipline of graph theory, the medial graph of plane graph G is another graph M(G) that represents the adjacencies between edges in the faces of G. Medial graphs were introduced in 1922 by Ernst Steinitz to study combinatorial properties of convex polyhedra, although the inverse construction was already used by Peter Tait in 1877 in his foundational study of knots and links.

Jon Hal Folkman was an American mathematician, a student of John Milnor, and a researcher at the RAND Corporation.

Paul Allen Catlin was a mathematician, professor of mathematics who worked in graph theory and number theory. He wrote a significant paper on the series of chromatic numbers and Brooks' theorem, titled Hajós graph coloring conjecture: variations and counterexamples.

<span class="mw-page-title-main">Kelmans–Seymour conjecture</span>

In graph theory, the Kelmans–Seymour conjecture states that every 5-vertex-connected graph that is not planar contains a subdivision of the 5-vertex complete graph K5. It is named for Paul Seymour and Alexander Kelmans, who independently described the conjecture; Seymour in 1977 and Kelmans in 1979. A proof was announced in 2016, and published in four papers in 2020.

<span class="mw-page-title-main">June Huh</span> American mathematician (born 1983)

June Huh is an American mathematician who is currently a professor at Princeton University. Previously, he was a professor at Stanford University. He was awarded the Fields Medal in 2022 and a MacArthur Fellowship in 2022. He has been noted for the linkages that he has found between algebraic geometry and combinatorics.

References

  1. 1 2 Baker, Matthew (January 2018). "Hodge theory in combinatorics". Bulletin of the American Mathematical Society. 55 (1): 57–80. doi: 10.1090/bull/1599 . ISSN   0273-0979. S2CID   51813455.
  2. R. C. Read, An introduction to chromatic polynomials, J. Combinatorial Theory 4 (1968), 52–71. MR0224505 (37:104)
  3. Hoggar, S. G (1974-06-01). "Chromatic polynomials and logarithmic concavity". Journal of Combinatorial Theory . Series B. 16 (3): 248–254. doi: 10.1016/0095-8956(74)90071-9 . ISSN   0095-8956.
  4. Huh, June. "Hard Lefschetz theorem and Hodge-Riemann relations for combinatorial geometries" (PDF).
  5. "He Dropped Out to Become a Poet. Now He's Won a Fields Medal". Quanta Magazine . 5 July 2022. Retrieved 5 July 2022.
  6. Kalai, Gil (July 2022). "The Work of June Huh" (PDF). Proceedings of the International Congress of Mathematicians 2022: 1–16., pp. 24.
  7. Huh, June (2012). "Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs". Journal of the American Mathematical Society. 25: 907–927. arXiv: 1008.4749 . doi: 10.1090/S0894-0347-2012-00731-0 .