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

The simple connected graphs may be classified by their cycle composition (number and lengths of cycles). This work derives the counting series of the simple connected graphs that have cycles of unrestricted number and length, but no…

Let $G=(V,E)$ be a graph and $p$ a positive integer. The $p$-domination number $\g_p(G)$ is the minimum cardinality of a set $D\subseteq V$ with $|N_G(x)\cap D|\geq p$ for all $x\in V\setminus D$. The $p$-reinforcement number $r_p(G)$ is…

A rooted arborescence of a directed graph is a spanning tree directed towards a particular vertex. A recent work of Chepuri et al. showed that the arborescences of a covering graph of a directed graph G are closely related to the…

Let $G$ be a finite group. For some fixed prime $p$, let $\Gamma_p(G)$ be the common divisor graph built on the set of sizes of $p$-regular conjugacy classes of $G$: this is the simple undirected graph whose vertices are the class sizes of…

A graph is odd if all of its vertices have odd degrees. In particular, an odd spanning tree in a connected graph is a spanning tree in which all vertices have odd degrees. In this paper we establish a unified technique to enumerate odd…

A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the same parity as $n$, if the removal of any set of $p$ vertices results in a graph with a perfect matching. 1-Factor-critical graphs and 2-factor-critical graphs are…

In this paper we show that every finite spatial graph is a connected sum of a planar graph, which is a forest, i.e. disjoint union of finite number of trees and a tangle. As a consequence we get that any finite spatial graph is a connected…

In this paper we show that prime sum graphs on $n$ vertices -- which are graphs on vertex set $\{1,2,...,n\}$ where $ij$ is an edge when $i+j$ is prime -- contain all trees with at most $\exp( c \log n / \log\log n)$ vertices as induced…

In their study of fundamental groups of one-dimensional path-connected compact metric spaces, Cannon and Conner have asked: Is there a tree-like object that might be considered the topological Cayley graph? We answer this question in the…

In this paper we establish all extremal graphs with respect to augmented eccentric connectivity index among all (simple connected) graphs, among trees and among trees with perfect matching. For graphs that turn out to be extremal explicit…

A graph G is said to be 1-perfectly orientable (1-p.o. for short) if it admits an orientation such that the out-neighborhood of every vertex is a clique in G. The class of 1-p.o. graphs forms a common generalization of the classes of…

A cyclic base ordering of a connected graph $G$, is a cyclic ordering of $E(G)$ such that every cyclically consecutive $|V(G)|-1$ edges form a spanning tree. In this project, we study cyclic base ordering of various families of graphs,…

We show that $(n,d,\lambda)$-graphs with $\lambda=O(d/\log^3 n)$ are universal with respect to all bounded degree spanning trees. This significantly improves upon the previous best bound due to Han and Yang of the form…

A $k$-tree is a spanning tree in which every vertex has degree at most $k$. In this paper, we provide a sufficient condition for the existence of a $k$-tree in a connected graph with fixed order in terms of the adjacency spectral radius and…

In this paper we consider the problem of connected edge searching of weighted trees. It is shown that there exists a polynomial-time algorithm for finding optimal connected search strategy for bounded degree trees with arbitrary weights on…

We define a total order, which we call rooted order, on minimal generating set of $J(P_n)^s$ where $J(P_n)$ is the cover ideal of a path graph on $n$ vertices. We show that each power of a cover ideal of a path has linear quotients with…

A rotor-router walk is a deterministic version of a random walk, in which the walker is routed to each of the neighbouring vertices in some fixed cyclic order. We consider here directed covers of graphs (called also periodic trees) and we…

Given a graph, we can form a spanning forest by first sorting the edges in some order, and then only keep edges incident to a vertex which is not incident to any previous edge. The resulting forest is dependent on the ordering of the edges,…

The center, median and the security center are three central parts defined for any connected graph whereas the characteristic set, subtree core and core vertices are three central parts defined for trees only. We extend the concept of the…

Disjoint-Set forests, consisting of Union-Find trees are data structures having a widespread practical application due to their efficiency. Despite them being well-known, no exact structural characterization of these trees is known (such a…