Related papers: Not all partial cubes are $\Theta$-graceful

We show that the second homology of the configuration spaces of a planar graph is generated under the operations of embedding, disjoint union, and edge stabilization by three atomic graphs: the cycle graph with one edge, the star graph with…

It is well known that a graph with $m$ edges can be made triangle-free by removing (slightly less than) $m/2$ edges. On the other hand, there are many classes of graphs which are hard to make triangle-free in the sense that it is necessary…

This note is about the Chow ring of Verra fourfolds. For a general Verra fourfold, we show that the Chow group of homologically trivial $1$-cycles is generated by conics. We also show that Verra fourfolds admit a multiplicative…

In this article bipartite planar graphs St_r are investigated, r the number of their plane regions. Bounds for the graded Betti numbers and the projective dimension of the quotient ring associated to such graphs are discussed. We prove that…

In this paper we define some new labellings for trees, called the in-improper and out-improper odd-graceful labellings such that some trees labelled with the new labellings can induce graceful graphs having at least a cycle. We, next, apply…

An unfriendly partition of a graph $G = (V,E)$ is a function $c: V \to 2$ such that $|\{x\in N(v): c(x)\neq c(v)\}|\geq |\{x\in N(v): c(x)=c(v)\}|$ for every vertex $v\in V$, where $N(v)$ denotes its neighborhood. It was conjectured by…

Let $G$ be a cyclically $5$-connected cubic graph with a $5$-edge-cut separating $G$ into two cyclic components $G_1$ and $G_2$. We prove that each component $G_i$ can be completed to a cyclically $5$-connected cubic graph by adding three…

When all ternary cubic forms over $\mathbb Z$ are ordered by the heights of their coefficients, we show that a positive proportion of them fail the Hasse principle, i.e., they have a zero over every completion of $\mathbb Q$ but no zero…

We show that the vertices of every planar graph can be partitioned into two sets, each inducing a so-called triangle-forest, i.e., a graph with no cycles of length more than three. We further discuss extensions to locally planar graphs.…

In this paper, we show that every planar graph without $4$-cycles and $6$-cycles has a partition of its vertex set into two sets, where one set induces a forest, and the other induces a forest with maximum degree at most $2$ (equivalently,…

Let the matching polynomial of a graph $G$ be denoted by $\mu (G,x)$. A graph $G$ is said to be $\theta$-super positive if $\mu(G,\theta)\neq 0$ and $\mu(G\setminus v,\theta)=0$ for all $v\in V(G)$. In particular, $G$ is 0-super positive if…

The subdivided double construction on 4-regular graphs was used by Poto\v{c}nik and Wilson to explore semi-symmetric (edge-transitive but not vertex-transitive) graphs, and can be used to construct every semi-symmetric 4-regular graph that…

We propose a Law of Nature? Viz., Pure Regularity Occurs at Na\"ive Levels and Regularity has Affinity with Evenness. In a series of three papers, it was established that regular Euler graphs with only one type of (pure) cycles are…

Tuza's Conjecture states that if a graph $G$ does not contain more than $k$ edge-disjoint triangles, then some set of at most $2k$ edges meets all triangles of $G$. We prove Tuza's Conjecture for all graphs $G$ having no subgraph with…

Xu and Wu proved that if every $5$-cycle of a planar graph $G$ is not simultaneously adjacent to $3$-cycles and $4$-cycles, then $G$ is $4$-choosable. In this paper, we improve this result as follows. Let $\{i, j, k, l\} = \{3,4,5,6\}.$ For…

We prove that there exist finitely generated, stably finite algebras which are non linear sofic. This was left open by Arzhantseva and P\u{a}unescu in 2017.

A 4-wheel is a graph formed by a cycle C and a vertex not in C that has at least four neighbors in C. We prove that a graph G that does not contain a 4-wheel as a subgraph is 4-colorable and we describe some structural properties of such a…

We extend the clique-coclique inequality, previously known to hold for graphs in association schemes and vertex-transitive graphs, to graphs in homogeneous coherent configurations and 1-walk regular graphs. We further generalize it to a…

We prove the existence of infinite classes of cyclic G-decompositions of the complete multipartite graph, G being a caterpillar, a hairy cycle or a cycle. All the results are obtained by the construction of d-divisible $\alpha$-labelings of…

Hibi conjectured that if a toric ideal has a quadratic Gr\"obner basis, then the toric ideal has either a lexicographic or a reverse lexicographic quadratic Gr\"obner basis. In this paper, we present a cut ideal of a graph that serves as a…