Related papers: New Classes of Set-Sequential Trees

The modular decomposition of a graph $G$ is a natural construction to capture key features of $G$ in terms of a labeled tree $(T,t)$ whose vertices are labeled as "series" ($1$), "parallel" ($0$) or "prime". However, full information of $G$…

A labeled oriented tree is called injective if each generator occurs at most once as an edge label. We show that injective labeled oriented trees are aspherical. The proof relies on a new relative asphericity test based on a lemma of…

For a graph $G$, the $\ell$-deck of $G$ is the multiset of induced subgraphs on $G$ having $\ell$ vertices. Recently, Groenland et al. proved that any tree can be reconstructed from its $(8/9+o(1))n$-deck. For the particular case of…

Labeled infinite trees provide combinatorial interpretations for many integer sequences generated by nested recurrence relations. Typically, such sequences are monotone increasing. Several of these sequences also have straightforward…

A graph $G$ is asymmetrizable if it has a set of vertices whose setwise stablizer only consists of the identity automorphism. The motion $m$ of a graph is the minimum number of vertices moved by any non-identity automorphism. It is known…

Many interesting problems are obtained by attempting to generalize classical results on convexity in Euclidean spaces to other convexity spaces, in particular to convexity spaces on graphs. In this paper we consider $P_3$-convexity on…

A $p$-caterpillar is a caterpillar such that every non-leaf vertex is adjacent to exactly $p$ leaves. We give a tight minimum degree condition for a graph to have a spanning $p$-caterpillar.

Let $G$ be a connected graph and $L(G)$ the set of all integers $k$ such that $G$ contains a spanning tree with exactly $k$ leaves. We show that for a connected graph $G$, the set $L(G)$ is contiguous. It follows from work of Chen, Ren, and…

A simple and connected $n$-vertex graph has a prime vertex labeling if the vertices can be injectively labeled with the integers $1, 2, 3,\ldots, n$, such that adjacent vertices have relatively prime labels. We will present previously…

A caterpillar graph is a tree which on removal of all its pendant vertices leaves a chordless path. The chordless path is called the backbone of the graph. The edges from the backbone to the pendant vertices are called the hairs of the…

Let $T$ be an oriented tree on $n$ vertices with maximum degree at most $e^{o(\sqrt{\log n})}$. If $G$ is a digraph on $n$ vertices with minimum semidegree $\delta^0(G)\geq(\frac12+o(1))n$, then $G$ contains $T$ as a spanning tree, as…

We call a tree $T$ is \emph{even} if every pair of its leaves is joined by a path of even length. Jackson and Yoshimoto~[J. Graph Theory, 2024] conjectured that every $r$-regular nonbipartite connected graph $G$ has a spanning even tree.…

Using the theory of Properly Embedded Graphs developed in an earlier work we define an involutory duality on the set labeled non-crossing trees that lifts the obvious duality in the set of unlabeled non-crossing trees. The set of…

We give a short and direct proof of a remarkable identity that arises in the enumeration of labeled trees with respect to their indegree sequence, where all edges are oriented from the vertex with lower label towards the vertex with higher…

A subtree of a tree is any induced subgraph that is again a tree (i.e., connected). The mean subtree order of a tree is the average number of vertices of its subtrees. This invariant was first analyzed in the 1980s by Jamison. An intriguing…

The Gy\'arf\'as tree packing conjecture asserts that any set of trees with $2,3, ..., k$ vertices has an (edge-disjoint) packing into the complete graph on $k$ vertices. Gy\'arf\'as and Lehel proved that the conjecture holds in some special…

The $(n-\ell)$-deck of an $n$-vertex graph is the multiset of subgraphs obtained from it by deleting $\ell$ vertices. A family of $n$-vertex graphs is $\ell$-recognizable if every graph having the same $(n-\ell)$-deck as a graph in the…

The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labelled by using the numbers {1,2,...,n} in such a way that the absolute differences induced on the edges are pairwise…

For a given graph, the unlabeled subgraphs $G-v$ are called the cards of $G$ and the deck of $G$ is the multiset $\{G-v: v \in V(G)\}$. Wendy Myrvold [Ars Combinatoria, 1989] showed that a non-connected graph and a connected graph both on…

Balister, the second author, Groenland, Johnston and Scott recently showed that there are asymptotically $C4^n/n^{3/4}$ many unordered sequences that occur as degree sequences of graphs. Combining limit theory for infinitely divisible…