Related papers: A Computational Approach to the Graceful Tree Conj…
We give a proof for sharp estimate for the number of spanning trees using linear algebra and generalize this bound to multigraphs. In addition, we show that this bound is tight for complete graphs. In addition, we give estimates for number…
A drawing of a given (abstract) tree that is a minimum spanning tree of the vertex set is considered aesthetically pleasing. However, such a drawing can only exist if the tree has maximum degree at most 6. What can be said for trees of…
Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with $k$ roots and that with $k+1$ roots on…
We prove the following sharp estimate for the number of spanning trees of a graph in terms of its vertex-degrees: a simple graph $G$ on $n$ vertices has at most $(1/n^{2}) \prod_{v \in V(G)} (d(v)+1)$ spanning trees. This result is tight…
We consider two varieties of labeled rooted trees, and the probability that a vertex chosen from all vertices of all trees of a given size uniformly at random has a given rank. We prove that this probability converges to a limit as the tree…
In this paper, we address the problem of packing large trees in $G_{n,p}$. In particular, we prove the following result. Suppose that $T_1, \dotsc, T_N$ are $n$-vertex trees, each of which has maximum degree at most $(np)^{1/6} / (\log…
An increasing 1,2-tree is a labeled graph formed by starting with a vertex and then repeatedly attaching a leaf to a vertex or a triangle to an edge, the labeling of the vertices corresponding to the order in which the vertices are added.…
The Unfriendly Partition Conjecture posits that every countable graph admits a 2-colouring in which for each vertex there are at least as many bichromatic edges containing that vertex as monochromatic ones. This is not known in general, but…
We consider the problem of finding a spanning tree with maximum number of leaves (MaxLeaf). A 2-approximation algorithm is known for this problem, and a 3/2-approximation algorithm when restricted to graphs where every vertex has degree 3…
In this paper, we study three algorithmic problems involving computation trees: the optimization, solvability, and satisfiability problems. The solvability problem is concerned with recognizing computation trees that solve problems. The…
We prove that finding a rooted subtree with at least $k$ leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family $\cal L$ that…
Speculative decoding is a promising approach for accelerating large language models. The primary idea is to use a lightweight draft model to speculate the output of the target model for multiple subsequent timesteps, and then verify them in…
Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $p$ and size $q$. For $k\ge 1$, a bijection $f: V(G)\cup E(G) \to \{k, k+1, k+2, \ldots, k+p+q-1\}$ such that $f(uv)= |f(u) - f(v)|$ for every edge $uv\in E(G)$ is said…
Characterizing graphs by their spectra is a fundamental and challenging problem in spectral graph theory, which has received considerable attention in recent years. A major unsolved conjecture in this area is Haemers' conjecture which…
In 1991, Gnanajothi [4] proved that the path graph P_n with n vertex and n-1 edge is odd graceful, and the cycle graph C_m with m vertex and m edges is odd graceful if and only if m even, she proved the cycle graph is not graceful if m odd.…
In 1991, Gnanajothi [4] proved that the path graph P_n with n vertex and n-1 edge is odd graceful, and the cycle graph C_m with m vertex and m edges is odd graceful if and only if m even, she proved the cycle graph is not graceful if m odd.…
We initiate the study of the hat guessing number of a graph where the adversary is only allowed to provide a proper coloring of the graph. This is the largest number $q$ for which there is a guessing strategy on each vertex that only…
Consider an order $n$ abelian group $G$ and a tree $T$ on $n$ vertices. When is it possible to (bijectively) label $V(T)$ by $G$ so that along all edges $xy$ of $T$, the sums $x+y$ are distinct? This problem can be traced back to the work…
Merge trees are fundamental structures in topological data analysis. Interleaving distance is a widely accepted metric for comparing merge trees, with applications in visualization and scientific computing. While a greedy algorithm exists…
We prove that there is $c>0$ such that for all sufficiently large $n$, if $T_1,\dots,T_n$ are any trees such that $T_i$ has $i$ vertices and maximum degree at most $cn/\log n$, then $\{T_1,\dots,T_n\}$ packs into $K_n$. Our main result…