Related papers: A splitter theorem on 3-connected matroids and gra…

A graph $G$ is said to be chordal if it has no induced cycles of length four or more. In a recent preprint Culbertson, Guralnik, and Stiller give a new characterization of chordal graphs in terms of sequences of what they call…

Stein proved that for each simple plane triangulation H there exists a partitioning of the vertex of H into two subsets each of which induces a forest if and only if the dual H^{*} has a Hamilton cycle. We extend the Stein theorem for…

Given a set $\mathcal{F}$ of graphs, we call a copy of a graph in $\mathcal{F}$ an $\mathcal{F}$-graph. The $\mathcal{F}$-isolation number of a graph $G$, denoted by $\iota(G,\mathcal{F})$, is the size of a smallest subset $D$ of the vertex…

Let $M$ be a 3-connected matroid, and let $N$ be a 3-connected minor of $M$. We say that a pair $\{x_1,x_2\} \subseteq E(M)$ is $N$-detachable if one of the matroids $M/x_1/x_2$ or $M \backslash x_1 \backslash x_2$ is both 3-connected and…

Let G be a graph with vertices V and edges E. Let F be the union-closed family of sets generated by E. Then F is the family of subsets of V without isolated points. Theorem: There is an edge e belongs to E such that |{U belongs to F | e…

A long-standing conjecture of Thomassen says that every longest cycle of a $3$-connected graph has a chord. Thomassen (2018) proved that if $G$ is $2$-connected and cubic, then any longest cycle must have a chord. He also showed that if $G$…

In this paper we study the operation of cutting off edges of a simple $3$-polytope $P$ along the graph $\Gamma$. We give the criterion when the resulting polytope is simple and when it is flag. As a corollary we prove the analog of…

Tutte has described in the book "Connectivity in graphs" a canonical decomposition of any graph into 3-connected components. In this article we translate (using the language of symbolic combinatorics) Tutte's decomposition into a general…

Let $n\geq 3$ and $r_n$ be a $3$-polytopal graph such that for every $3\leq i\leq n$, $r_n$ has at least one vertex of degree $i$. We find the minimal vertex count for $r_n$. We then describe an algorithm to construct the graphs $r_n$. A…

Let $G$ be a graph such that, whenever two vertices $x$ and $y$ of $G$ are joined by three internally disjoint paths, $x$ and $y$ are adjacent. Jamison and Mulder determined that the set of such graphs coincides with the set of graphs that…

We prove the following 30-year old conjecture of Gy\H{o}ri and Tuza: the edges of every $n$-vertex graph $G$ can be decomposed into complete graphs $C_1,\ldots,C_\ell$ of orders two and three such that $|C_1|+\cdots+|C_\ell|\le…

Let $S$ be a set of $n$ points in the plane in general position. Two line segments connecting pairs of points of $S$ cross if they have an interior point in common. Two vertex disjoint geometric graphs with vertices in $S$ cross if there…

Let $n\geq 6,k\geq 0$ be two integers. Let $H$ be a graph of order $n$ with $k$ components, each of which is an even cycle of length at least $6$ and $G$ be a bipartite graph with bipartition $(X,Y)$ such that $|X|=|Y|\geq n/2$. In this…

Let $M$ be a 3-connected binary matroid and let $Y(M)$ be the set of elements of $M$ avoiding at least $r(M)+1$ non-separating cocircuits of $M$. Lemos proved that $M$ is non-graphic if and only if $Y(M)\neq\emp$. We generalize this result…

Let D be a simple digraph without loops or digons. For any v in V(D) let N_1(v) be the set of all nodes at out-distance 1 from v and let N_2(v) be the set of all nodes at out-distance 2. We provide sufficient conditions under which there…

A theorem of Tutte states that every 4-connected non-hamiltonian graph contains $K_{3,3}$ as a minor. We strengthen this result by proving that such a graph must contain $K_{3,4}$ as a minor, thereby confirming a special case of a…

We characterize 2-dimensional complexes associated canonically with basis graphs of matroids as simply connected triangle-square complexes satisfying some local conditions. This proves a version of a (disproved) conjecture by Stephen Maurer…

We prove that if $G$ is a 2-connected graph with $\delta(G) \geqslant \frac{v(G) + 2}{3}$ then $G$ has a cycle $W$ such that $V(G - W)$ is independent. This result is best possible in the sense that it becomes false if $\frac{v(G) + 2}{3}$…

For a non-negative integer $s\le |V(G)|-3$, a graph $G$ is $s$-Hamiltonian if the removal of any $k\le s$ vertices results in a Hamiltonian graph. Given a connected simple graph $G$ that is not isomorphic to a path, a cycle, or a $K_{1,3}$,…

Recently, Zhang and Wu proved a conjecture of Kalai and Meshulam, showing that for every graph $G$ without induced cycles of length divisible by $3$, the sum of all reduced Betti numbers of its independence complex $I(G)$ is at most $1$. We…