Problems

Is the sequence W_0, W_1, …, W_n counting the flats of each rank of a matroid always unimodal? Rota conjectured yes in 1970.

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

Mason conjectured the following: let M be a matroid of rank r, and let W_i denote the number of flats of M of rank i. Is it true that for all 1 ≤ i ≤ r - 1, we have W_i^2 ≥ W_i + 1W_i - 1? This is false; a counterexample is given by a…

Erdos and Szemeredi conjectured that every finite set of reals satisfies max(|A+A|,|AA|) ≥ |A|^2-o(1). False: there are arbitrarily large A ⊂ R, of algebraic integers in a number field of degree asymp log|A|, with max(|A+A|,|AA|) ≤ |A|^2-c…

Let T_k be the least t such that every equinumerous t-coloring of [tn] contains a rainbow k-term arithmetic progression. Jungic, Licht, Mahdian, Nesetril and Radoicic conjectured T_k = Θ(k^2); Conlon, Fox and Sudakov proved T_k = O(k^2 log…

Akbari, Alikhani, Oboudi and Peng conjectured in 2010 that 0 and -2 are the only integer roots of the domination polynomial D(G, x), proven for trees and unicyclic graphs and verified exhaustively for small orders. The paper gives a…

WOW-284 asserts that the minimum dual degree of every connected graph of order at least three and girth at least five is at most the negative of its least distance eigenvalue. The paper refutes it with exact counterexamples of orders 38,…

For every n ≥ 2k + 1, is the independence polynomial of GP(n, k) real-rooted if and only if k is even? Exact Sturm counts refute both directions.

Campbell, Distel, Gollin, Harvey, Hendrey, Hickingbotham, Mohar and Wood conjectured that every graph of degree-d polynomial growth embeds into the strong product of d trees of linear growth and a bounded clique. False for d = 4: a…

White conjectured that the symmetric exchange binomials generate the toric ideal of a matroid. This is now known to be false; a rank 9 binary matroid constitutes a counterexample.

Hamaker and Reiner conjectured that the order complex of an open interval (u,w) in the ASM weak order is contractible unless w is the long element of a standard parabolic subgroup, in which case it is homotopy equivalent to a sphere.…

A graph G is maximal non-Hamiltonian if it is non-Hamiltonian but G + e is Hamiltonian for every nonedge e. In 1994 Vu Dinh Hoa conjectured a property of G - V(C) for a longest cycle C of such a graph. Disproved by an explicit base graph…

A cyclic coloration of a triangulation of a closed 2-manifold gives the faces around every vertex distinct colors. Chen and Lawrencenko made two conjectures about the cyclic chromatic number of minimal triangulations in 1999. Their second…

For every connected graph G, is α(G) ≤ ⌊ b(G) - log(ecc_avg(G)) ⌋, where b(G) is the largest induced-bipartite-subgraph order? An 11-vertex counterexample - a triangle with four leaves on each of two vertices - has α = 9 against bound 8.

For a graph G, its vertex deck is the multiset of graphs obtained by deleting one vertex. Bowler, Brown, and Fenner (BBF) proposed 2⌊(n−1)/3⌋ as the maximum possible overlap between the decks of two nonisomorphic n-vertex graphs, for all…

Albertson and Berman conjectured that for every simple planar graph G on n vertices, the largest vertex set inducing a forest has size at least n/2. The standing lower bound since the same year has been Borodin's 2n/5, from his acyclic…

Jaeger conjectured that every bridgeless cubic graph G admits a Petersen coloring: a map φcolon E(G)→ E(P) into the edges of the Petersen graph P such that, for every vertex v of G, the three edges at v are sent to three edges meeting at a…

Simonovits conjectured that if a forbidden family F with p(F) > 1 has extremal number exceeding the Turan bound by a superlinear surplus, then its extremal graphs are joins of p graphs, each extremal for a family of chromatic number two.…

A graph on n vertices is very well-covered if every maximal independent set has size n/2. Levit and Mandrescu conjectured that the independence polynomial i(G,x) of every very well-covered graph is unimodal, i.e. its coefficient sequence…

Teschner conjectured that every finite simple graph G with at least one edge satisfies b(G) ≤ 3/2Δ(G), where b(G) is the bondage number and Δ(G) is the maximum degree. Yavari gives a connected cubic bipartite graph on 18 vertices with…