Related papers: A counterexample to a result on the tree graph of …

The families EPT (resp. EPG) Edge Intersection Graphs of Paths in a tree (resp. in a grid) are well studied graph classes. Recently we introduced the graph classes Edge-Intersecting and Non-Splitting Paths in a Tree ENPT, and in a Grid…

A graph $G=(V,E)$ is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree $T$ and two non-negative real numbers $d_{min}$ and $d_{max}$, $d_{min} \leq d_{max}$, such that each node $u \in V$ is uniquely associated to a…

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex (edge) labeling with $d$ labels that is preserved only by the trivial automorphism. It is known that for every graph $G$…

For a graph $G$, $\chi(G)$ denotes the chromatic number of $G$ and $\omega(G)$ denotes the size of the largest clique in $G$. A hereditary class of graphs is called $\chi$-bounded if there is a function $f$ such that for each graph $G$ in…

For a given simple graph $G$, the energy of $G$, denoted by $\mathcal {E}(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix, which was defined by I. Gutman. The problem on determining the maximal…

In this paper, we prove the following conjecture proposed by Gould, Hirohata and Keller [Discrete Math. submitted]: Let $G$ be a graph of sufficiently large order. If $\sigma_t(G) \geq 2kt - t + 1$ for any two integers $k \geq 2$ and $t…

A graph $G=(V,E)$ is called a pairwise compatibility graph (PCG) if there exists an edge-weighted tree $T$ and two non-negative real numbers $d_{min}$ and $d_{max}$ such that each leaf $u$ of $T$ corresponds to a vertex $u \in V$ and there…

We consider questions related to the existence of spanning trees in graphs with the property that after the removal of any path in the tree the graph remains connected. We show that, for planar graphs, the existence of trees with this…

We show that for any fixed dense graph G and bounded-degree tree T on the same number of vertices, a modest random perturbation of G will typically contain a copy of T . This combines the viewpoints of the well-studied problems of embedding…

The number of spanning trees of a graph $G$, denoted $\tau(G)$, is a well studied graph parameter with numerous connections to other areas of mathematics. In a recent remarkable paper, answering a question of Sedl\'a\v{c}ek from 1969, Chan,…

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.

The transmission of a vertex $v$ of a graph $G$ is the sum of distances from $v$ to all the other vertices in $G$. A graph is transmission irregular if all of its vertices have pairwise different transmissions. A starlike tree…

Let $F=\{H_1,...,H_k\}$ be a family of graphs. A graph $G$ with $m$ edges is called {\em totally $F$-decomposable} if for {\em every} linear combination of the form $\alpha_1 e(H_1) + ... + \alpha_k e(H_k) = m$ where each $\alpha_i$ is a…

We consider the problem of uniformly generating a spanning tree, of a connected undirected graph. This process is useful to compute statistics, namely for phylogenetic trees. We describe a Markov chain for producing these trees. For cycle…

Let A_1,...,A_k be a collection of families of subsets of an n-element set. We say that this collection is cross-intersecting if for any i,j in [k] with i not equal to j, A in A_i and B in A_j implies that the intersection of A and B is…

The mean subtree order of a given graph $G$, denoted $\mu(G)$, is the average number of vertices in a subtree of $G$. Let $G$ be a connected graph. Chin, Gordon, MacPhee, and Vincent [J. Graph Theory, 89(4): 413-438, 2018] conjectured that…

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…

The transmission of a vertex in a connected graph is the sum of its distances to all the other vertices. A graph is transmission irregular (TI) when all of its vertices have mutually distinct transmissions. In an earlier paper, Al-Yakoob…

Given a bridgeless graph $G$, the Cycle Double Cover Conjecture posits that there is a list of cycles of $G$, such that every edge appears in exactly two cycles on the list. This conjecture was originally posed independently in 1973 by…

In this paper we investigate results of the form "every graph $G$ has a cycle $C$ such that the induced subgraph of $G$ on $V(G)\setminus V(C)$ has small maximum degree." Such results haven't been studied before, but are motivated by the…