Related papers: New graceful diameter-6 trees by transfers

Recently L. B. Beasley introduced $(2,3)$-cordial labelings of directed graphs in [1]. He made two conjectures which we resolve in this article. He conjectured that every orientation of a path of length at least five is $(2,3)$ cordial, and…

Let $G$ be an undirected simple connected graph. We say a vertex $u$ is eccentric to a vertex $v$ in $G$ if $d(u,v)=\max\{d(v,w): w\in V(G)\}$. The eccentric graph, $E(G)$ of $G$ is a graph defined on the same vertex set as of $G$ and two…

In "On the super edge graceful trees of even orders," Chung, Lee, Gao, and Schaffer posed the following problem: Characterize trees of diameter 4 which are super edge-graceful. In this paper, we provide super edge-graceful labelings for all…

A rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph…

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…

A spanning subgraph $F$ of a graph $G$ is called perfect if $F$ is a forest, the degree $d_F(x)$ of each vertex $x$ in $F$ is odd, and each tree of $F$ is an induced subgraph of $G$. We provide a short proof of the following theorem of A.D.…

Merge trees, contour trees, and Reeb graphs are graph-based topological descriptors that capture topological changes of (sub)level sets of scalar fields. Comparing scalar fields using their topological descriptors has many applications in…

A labeling of the vertices of a graph by elements of any abelian group $A$ induces a labeling of the edges by summing the labels of their endpoints. Hovey defined the graph $G$ to be $A$-cordial if it has such a labeling where the vertex…

Given a graph $G$, the family of all independent sets of size $k$ containing a fixed vertex $v$ is called a star with centre $v$, and is denoted by $\mathcal{I}_G^k(v)$. Motivated by a generalisation of the Erd\H{o}s-Ko-Rado Theorem to the…

A graph $G$ is equitably $k$-list arborable if for any $k$-uniform list assignment $L$, there is an equitable $L$-colouring of $G$ whose each colour class induces an acyclic graph. The smallest number $k$ admitting such a coloring is named…

A vertex $v$ is called an AR-vertex, if $v$ has distinct edge weight sums for each distinct subset of edges incident on $v$. i.e., if $\{x_1,x_2,\dots,x_k\}$ are the edge labels of the edges incident on $v$, then the $2^k$ subset sums are…

A signed graph $G$ is a graph where each edge is assigned a + (positive edge) or a - (negative edge). The signed degree of a vertex $v$ in a signed graph, denoted by $sdeg(v)$, is the number of positive edges incident to $v$ subtracted by…

A graph $G$ is cordial if there exists a function $f$ from the vertices of $G$ to $\{0,1\}$ such that the number of vertices labelled $0$ and the number of vertices labelled $1$ differ by at most $1$, and if we assign to each edge $xy$ the…

If $A$ is a finite Abelian group, then a labeling $f \colon E (G) \rightarrow A$ of the edges of some graph $G$ induces a vertex labeling on $G$; the vertex $u$ receives the label $\sum_{v\in N(u)}f (v)$, where $N(u)$ is an open…

For a labelled tree on the vertex set $[n]:=\{1,2,..., n\}$, define the direction of each edge $ij$ to be $i\to j$ if $i<j$. The indegree sequence of $T$ can be considered as a partition $\lambda \vdash n-1$. The enumeration of trees with a…

Wu, Zhang and Li [4] conjectured that the set of vertices of any simple graph $G$ can be equitably partitioned into $\lceil(\Delta(G)+1)/2\rceil$ subsets so that each of them induces a forest of $G$. In this note, we prove this conjecture…

Rough graph is the graphical structure of information system with imprecise knowledge. Tong He designed the properties of rough graph in 2006[6] and following that He and Shi introduced the notion of edge rough graph[7]. He et al developed…

We show the Graceful Tree Conjecture holds.

A graph labeling assigns values to the components of a graph (vertices, edges, etc.). In particular, distance magic labelings have been widely studied in undirected graphs. In such a labeling, the vertices are labeled with unique values…

Given a graph $G$ rooted at a vertex $r$ and weight functions, $\gamma, \tau: E(G) \rightarrow \mathbb{R}$, the generalized cable-trench problem (CTP) is to find a single spanning tree that simultaneously minimizes the sum of the total edge…