Tutte proved that every 3-connected graph on more than 4 nodes has a contractible edge. Barnette and Grünbaum proved the existence of a removable edge in the same setting. We show that the sequence of contractions and the sequence of removals from G to the K4 can be computed in O(|V | ) time by extending Barnette and Grünbaum’s theorem. As an application, we derive a certificate for the 3-conne...

We prove that for d ≥ 3, the 1-skeleton of any (d− 1)-dimensional doubly Cohen-Macaulay (abbreviated 2-CM) complex is generically drigid. This implies that Barnette’s lower bound inequalities for boundary complexes of simplicial polytopes ([4],[3]) hold for every 2-CM complex of dimension ≥ 2 (see Kalai [8]). Moreover, the initial part (g0, g1, g2) of the g-vector of a 2-CM complex (of dimensio...

Suppose σ ∈ Lω1,ω(L) is such that all equations occurring in σ are positive, have the same set of variables on each side of the equality symbol, and have at least one function symbol on each side of the equality symbol. We show that σ satisfies Vaught’s conjecture. In particular this proves Vaught’s conjecture for sentences of Lω1,ω(L) without equality.

The chromatic number of the categorical product of two n-tournaments can be strictly smaller than n. We show that min{χ(S × T ) : S and T are n-tournaments} is asymptotically equal to λn, where 12 ≤ λ ≤ 2 3 .

We develop a theory for the Hermite-Krichever Ansatz on the Heun equation. As a byproduct, we find formulae which reduce hyperelliptic integrals to elliptic ones.

Analogues and variations of Ulam’s notorious graph reconstruction conjecture have been studied for a variety of combinatorial objects, for instance words (see Schützenberger and Simon [2, Theorem 6.2.16]), permutations (see Raykova [4] and Smith [5]), and compositions (see Vatter [6]), to name a few. In answer to Cameron’s query [1] about the partition context, Pretzel and Siemons [3] proved th...

‎an  oriented perfect path double cover (oppdc) of a‎ ‎graph $g$ is a collection of directed paths in the symmetric‎ ‎orientation $g_s$ of‎ ‎$g$ such that‎ ‎each arc‎ ‎of $g_s$ lies in exactly one of the paths and each‎ ‎vertex of $g$ appears just once as a beginning and just once as an‎ ‎end of a path‎. ‎maxov{'a} and ne{v{s}}et{v{r}}il (discrete‎ ‎math‎. ‎276 (2004) 287-294) conjectured that ...

Around 1960, Markus & Yamabe [1] conjectured that, for each n ≥ 2, every rest point x0 of a nonlinear, class C1, n-dimensional system of differential equations ẋ = F (x) is globally asymptotically stable if the eigenvalues of the jacobian matrix F (x) have negative real parts at every point x in R. At that time, Olech [2] connected this problem with the question of injectivity of the mapping x ...

A graph is diameter two edge-critical if its diameter is two and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter two edge-critical graph on n vertices is at most ⌊ n2 4 ⌋ and the extremal graph is the complete bipartite graph Kb 2 c,d 2 e. In the series papers [8–10], the Murty-Simon Conjecture stated by Haynes et al. is not th...

A distinguishing r-labeling of a digraph G is a mapping λ from the set of vertices of G to the set of labels {1, . . . , r} such that no nontrivial automorphism of G preserves all the labels. The distinguishing number D(G) of G is then the smallest r for which G admits a distinguishing r-labeling. From a result of Gluck (David Gluck, Trivial set-stabilizers in finite permutation groups, Can. J....