Related papers: Strong log-convexity of genus sequences

We formulate conditions on a set of log-concave sequences, under which any linear combination of those sequences is log-concave, and further, of conditions under which linear combinations of log-concave sequences that have been transformed…

Given a graph $G$, its genus polynomial is $\Gamma_G(x) = \sum_{k\geq 0} g_k(G)x^k$, where $g_k(G)$ is the number of 2-cell embeddings of $G$ in an orientable surface of genus $k$. The Log-Concavity Genus Distribution (LCGD) Conjecture…

A "folklore conjecture, probably due to Tutte" (as described in [P.D. Seymour, Sums of circuits, Graph theory and related topics (Proc. Conf., Univ. Waterloo, 1977), pp. 341-355, Academic Press, 1979]) asserts that every bridgeless cubic…

Loebl, Koml\'os and S\'os conjectured that every $n$-vertex graph $G$ with at least $n/2$ vertices of degree at least $k$ contains each tree $T$ of order $k+1$ as a subgraph. We give a sketch of a proof of the approximate version of this…

Let (W, S) be a Coxeter system. A W-graph is an encoding of a representation of the corresponding Iwahori-Hecke algebra. Especially important examples include the W-graph corresponding to the action of the Iwahori-Hecke algebra on the…

This is the second of a series of four papers in which we prove the following relaxation of the Loebl-Komlos--Sos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at…

Wu, Zhang and Li [4] conjectured that the set of vertices of any simple graph $G$ can be equitably partitioned into $\lceil(\Delta(G)+1)/2\rceil$ subsets so that each of them induces a forest of $G$. In this note, we prove this conjecture…

The $2$-cell embeddings of graphs on closed surfaces have been widely studied. It is well known that ($2$-cell) embedding a given graph $G$ on a closed orientable surface is equivalent to cyclically ordering the edges incident to each…

Given a graph $G$, its independence sequence is the integral sequence $a_1,a_2,...,a_n$, where $a_i$ is the number of independent sets of vertices of size i. In the late 80's Alavi, Erdos, Malde, Schwenk showed that this sequence need not…

This is the third of a series of four papers in which we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at…

In a recent paper, the first author proved the log-concavity of the coefficients of the characteristic polynomial of a matroid realizable over a field of characteristic 0, answering a long-standing conjecture of Read in graph theory. We…

Gross, Mansour and Tucker introduced the partial-dual orientable genus polynomial and the partial-dual Euler genus polynomial. They computed these two partial-dual genus polynomials of four families of ribbon graphs, posed some research…

In this paper we define and study a new family of graphs that generalises the notions of line graphs and path graphs. Let $G$ be a graph with no loops but possibly with parallel edges. An \emph{$\ell$-link} of $G$ is a walk of $G$ of length…

Recently, the bunkbed conjecture has been shown to be false, which naturally prompts questions on how to classify the graphs that still satisfy the conjecture. We distinguish between a weak version of the bunkbed conjecture where all the…

The Rota--Heron--Welsh conjecture (now a theorem of Adiprasito, Huh, and the author) asserts the log-concavity of the characteristic polynomial of matroids. We give an exposition of the Lorentzian polynomial proof following the work of…

The linear arboricity of a graph $G$, denoted by $\text{la}(G)$, is the minimum number of edge-disjoint linear forests (i.e. forests in which every connected component is a path) in $G$ whose union covers all the edges of $G$. A famous…

The Linear Arboricity Conjecture asserts that the linear arboricity of a graph with maximum degree $\Delta$ is $\lceil (\Delta+1)/2 \rceil$. For a $2k$-regular graph $G$, this implies $la(G) = k+1$. In this note, we utilize a network flow…

The purpose of this note is to explain that the combinatorial local log-concavity conjecture introduced by Gross, Mansour, Tucker and Wang (Eur. J. Comb. 52, 207-222, 2016) in fact follows from a result of Stanley (Eur. J. Comb. 32 (6),…

This is the last paper of a series of four papers in which we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number~$k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at…

Anari, Gharan, and Vinzant proved (complete) log-concavity of the basis generating functions for all matroids. From the viewpoint of combinatorial Hodge theory, it is natural to ask whether these functions are "strictly" log-concave for…