Related papers: Generalizing $p$-goodness to ordered graphs

We consider the class of graphs for which the edge connectivity is equal to the maximum number of edge-disjoint spanning trees, and the natural generalization to matroids, where the cogirth is equal to the number of disjoint bases. We…

Working with tree graphs is always easier than with loopy ones and spanning trees are the closest tree-like structures to a given graph. We find a correspondence between the solutions of random K-satisfiability problem and those of spanning…

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…

In this paper, we find a criterium for universal equivalence of partially commutative Lie algebras whose defining graphs are trees. Besides, we obtain bases for partially commutative metabelian Lie algebras.

We show that every connected graph has a spanning tree that displays all its topological ends. This proves a 1964 conjecture of Halin in corrected form, and settles a problem of Diestel from 1992.

Consider the following problem: Given a planar graph $G$, what is the maximum number $p$ such that $G$ has a planar straight-line drawing with $p$ collinear vertices? This problem resides at the core of several graph drawing problems,…

Let $\triangleleft$ be a relation between graphs. We say a graph $G$ is \emph{$\triangleleft$-ubiquitous} if whenever $\Gamma$ is a graph with $nG \triangleleft \Gamma$ for all $n \in \mathbb{N}$, then one also has $\aleph_0 G \triangleleft…

A total weighting of a graph $G$ is a mapping $\phi$ that assigns a weight to each vertex and each edge of $G$. The vertex-sum of $v \in V(G)$ with respect to $\phi$ is $S_{\phi}(v)=\sum_{e\in E(v)}\phi(e)+\phi(v)$. A total weighting is…

Ordered matchings, defined as graphs with linearly ordered vertices, where each vertex is connected to exactly one edge, play a crucial role in the area of ordered graphs and their homomorphisms. Therefore, we consider related problems from…

A {\it tiered graph} $G=(V,E)$ with $m $ tiers is a simple graph with $V\subseteq \brk{n}$, where $\brk{n}=\{1,2,\cdots,n\}$, and with a surjective map $t$ from $V$ to $\brk{m}$ such that if $v$ is a vertex adjacent to $v'$ in $G$ with…

In the aftermath of the Robertson--Seymour Graph Minor Theorem, Thomas conjectured that the countable graphs are well-quasi-ordered under the minor relation. We prove that this conjecture, when restricted to graphs with no infinite paths…

Let $G$ be a 3-connected planar graph. Define the co-tree of a spanning tree $T$ of $G$ as the graph induced by the dual edges of $E(G)-E(T)$. The well-known cut-cycle duality implies that the co-tree is itself a tree. Let a $k$-tree be a…

Let $G$ be a graph of order $n$. The path decomposition of $G$ is a set of disjoint paths, say $\mathcal{P}$, which cover all vertices of $G$. If all paths are induced paths in $G$, then we say $\mathcal{P}$ is an induced path decomposition…

We introduce a class of rings, namely the class of left or right $p$-nil rings, for which the adjoint groups behave regularly. Every $p$-ring is close to being left or right $p$-nil in the sense that it contains a large ideal belonging to…

Given an undirected graph $G$, let us randomly orient $G$ by tossing independent (possibly biased) coins, one for each edge of $G$. Writing $a\rightarrow b$ for the event that there exists a directed path from a vertex $a$ to a vertex $b$…

An automorphism of a group is called outer if it is not an inner automorphism. Let $G$ be a finite $p$-group. Then for every outer $p$-automorphism $\phi$ of $G$ the subgroup $C_G(\phi)=\{x\in G \;|\; x^\phi=x\}$ has order $p$ if and only…

For any integer $n$, we classify all trees whose $n$-path ideals have linear quotients.

Let $G$ be a finite group. A number of graphs with the vertex set $G$ have been studied, including the power graph, enhanced power graph, and commuting graph. These graphs form a hierarchy under the inclusion of edge sets, and it is useful…

In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of…

The following theorem is proved: For all $k$-connected graphs $G$ and $H$ each with at least $n$ vertices, the treewidth of the cartesian product of $G$ and $H$ is at least $k(n -2k+2)-1$. For $n\gg k$ this lower bound is asymptotically…