相关论文: (2,3) Cordial Trees and Paths
In graph theory, a graceful labeling of a graph with m edges is a labeling of its vertices with a subset of the integers ranging from 0 to m inclusive, such that no two vertices share a label, and each edge is uniquely identified by the…
A variant of the Erd\H{o}s-S\'os conjecture, posed by Havet, Reed, Stein and Wood, states that every graph with minimum degree at least $\lfloor 2k/3 \rfloor$ and maximum degree at least $k$ contains a copy of every tree with $k$ edges.…
A graph drawing is $\textit{greedy}$ if, for every ordered pair of vertices $(x,y)$, there is a path from $x$ to $y$ such that the Euclidean distance to $y$ decreases monotonically at every vertex of the path. Greedy drawings support a…
Alon and Yuster proved that the number of orientations of any $n$-vertex graph in which every $K_3$ is transitively oriented is at most $2^{\lfloor n^2/4\rfloor}$ for $n \geq 10^4$ and conjectured that the precise lower bound on $n$ should…
Degree ssortativity is the tendency for nodes of high degree (resp.low degree) in a graph to be connected to high degree nodes (resp. to low degree ones). It is sually quantified by the Pearson correlation coefficient of the degree-degree…
Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case in which such graphs are Cayley graphs of Abelian groups. These groups can be constructed by…
We prove that for every $k$ and every $\varepsilon>0$, there exists $g$ such that every graph with tree-width at most $k$ and odd-girth at least $g$ has circular chromatic number at most $2+\varepsilon$.
An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. These codes have been widely studied for over two decades. We give an improvement…
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 show that if $D$ is an $n$-vertex digraph with more than $(k-1)n$ arcs that does not contain any of three forbidden digraphs, then $D$ contains every antidirected tree on $k$ arcs. The forbidden digraphs are those orientations of $K_{2,…
For a labeled tree on the vertex set $\set{1,2,\ldots,n}$, the local direction of each edge $(i\,j)$ is from $i$ to $j$ if $i<j$. For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges…
In this paper, we address the Ehrenborg's conjecture which proposes that for any bipartite graph the number of spanning trees does not exceed the product of the degrees of the vertices divided by the product of the sizes of the graph…
Mixed graphs can be seen as digraphs with arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite and in which the undirected and directed degrees are one. The best graphs,…
One of Erdos's conjectures states that every triangle-free graph on $n$ vertices has an induced subgraph on $n/2$ vertices with at most $n^2/50$ edges. We report several partial results towards this conjecture. In particular, we establish…
In this note we confirm a conjecture raised by Benjamini et al. \cite{BST} on the acquaintance time of graphs, proving that for all graphs $G$ with $n$ vertices it holds that $\AC(G) = O(n^{3/2})$, which is tight up to a multiplicative…
The girth of a graph $G$ is the length of a shortest cycle of $G$. Jiang (JCT-B, 2001) showed that every graph $G$ with girth at least $2\ell+1$ and minimum degree at least $k/\ell$ contains every tree $T$ with $k$ edges whose maximum…
The Wiener index of a connected graph is defined as the sum of distances between all its unordered pairs of vertices. Characterising graphs on $n$ vertices with a fixed diameter that maximise the Wiener index is a long-standing open…
Accessibility percolation is a new type of percolation problem inspired by evolutionary biology. To each vertex of a graph a random number is assigned and a path through the graph is called accessible if all numbers along the path are in…
Luo, Tian and Wu (2022) conjectured that for any tree $T$ with bipartition $X$ and $Y$, every $k$-connected bipartite graph $G$ with minimum degree at least $k+t$, where $t=$max$\{|X|,|Y|\}$, contains a tree $T'\cong T$ such that $G-V(T')$…
We study "positive" graphs that have a nonnegative homomorphism number into every edge-weighted graph (where the edgeweights may be negative). We conjecture that all positive graphs can be obtained by taking two copies of an arbitrary…