Related papers: Invitation to the subpath number

Consider a setting where possibly sensitive information sent over a path in a network is visible to every {neighbor} of the path, i.e., every neighbor of some node on the path, thus including the nodes on the path itself. The exposure of a…

Completely independent spanning trees in a graph $G$ are spanning trees of $G$ such that for any two distinct vertices of $G$, the paths between them in the spanning trees are pairwise edge-disjoint and internally vertex-disjoint. In this…

A supergrid graph is a finite vertex-induced subgraph of the infinite graph whose vertex set consists of all points of the plane with integer coordinates and in which two vertices are adjacent if the difference of their x or y coordinates…

Given a graph $G$, the maximal induced subgraphs problem asks to enumerate all maximal induced subgraphs of $G$ that belong to a certain hereditary graph class. While its optimization version, known as the minimum vertex deletion problem in…

A graph $G$ is said to be the intersection of graphs $G_1,G_2,\ldots,G_k$ if $V(G)=V(G_1)=V(G_2)=\cdots=V(G_k)$ and $E(G)=E(G_1)\cap E(G_2)\cap\cdots\cap E(G_k)$. For a graph $G$, $\mathrm{dim}_{COG}(G)$ (resp. $\mathrm{dim}_{TH}(G)$)…

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…

In this paper we study cycles in random bipartite graph $G(n,n,p)$. We prove that if $p\gg n^{-2/3}$, then $G(n,n,p)$ a.a.s. satisfies the following. Every subgraph $G'\subset G(n,n,p)$ with more than $(1+o(1))n^2p/2$ edges contains a cycle…

For a finite group $G$, we define the inclusion graph of subgroups of $G$, denoted by $\mathcal I(G)$, is a graph having all the proper subgroups of $G$ as its vertices and two distinct vertices $H$ and $K$ in $\mathcal I(G)$ are adjacent…

Let H be a graph, and let C_H(G) be the number of (subgraph isomorphic) copies of H contained in a graph G. We investigate the fundamental problem of estimating C_H(G). Previous results cover only a few specific instances of this general…

A (3,4)-biregular bigraph G is a bipartite graph where all vertices in one part have degree 3 and all vertices in the other part have degree 4. A path factor of G is a spanning subgraph whose components are nontrivial paths. We prove that a…

A longest path in a graph is called a detour. It is easy to see that a connected graph of minimum degree at least $2$ and order at least $4$ has at least $4$ detours. We prove that if the number of detours in such a graph of order at least…

For a $2$-connected graph $G$ and vertices $u,v$ of $G$ we define an abstract graph $\mathcal{P}(G_{uv})$ whose vertices are the paths joining $u$ and $v$ in $G$, where paths $S$ and $T$ are adjacent if $T$ is obtained from $S$ by replacing…

Let $G$ be a graph of order $n$. The path decomposition of $G$ is a set of disjoint paths, say $\mathcal{P}$, which cover all vertices of $G$. If all paths are induced paths in $G$, then we say $\mathcal{P}$ is an induced path decomposition…

An edge-colored graph $G$ is \emph{conflict-free connected} if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The \emph{conflict-free connection number} of a connected graph $G$,…

In 2012, Mader conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with minimum degree at least $\lfloor \frac{3k}{2}\rfloor+m-1$ contains a subtree $T'\cong T$ such that $G-V(T')$ remains $k$-connected. In 2022,…

Let $G$ be a simple graph with order $n$ and adjacency matrix $\mathbf{A}(G)$. Let $\phi(G; \lambda)=\det(\lambda I-\mathbf{A}(G))=\sum_{i=0}^n\mathbf{a}_i(G)\lambda^{n-i}$ be the characteristic polynomial of $G$, where $\mathbf{a}_i(G)$ is…

Subgraph enumeration problems ask to output all subgraphs of an input graph that belongs to the specified graph class or satisfy the given constraint. These problems have been widely studied in theoretical computer science. As far, many…

For a class $\mathcal{G}$ of graphs, the problem SUBGRAPH COMPLEMENT TO $\mathcal{G}$ asks whether one can find a subset $S$ of vertices of the input graph $G$ such that complementing the subgraph induced by $S$ in $G$ results in a graph in…

Using the theory of electrical network, we first obtain a simple formula for the number of spanning trees of a complete bipartite graph containing a certain matching or a certain tree. Then we apply the effective resistance (i.e.,…

Let $\mathcal{G}$ be the set of simple graphs (or multigraphs) $G$ such that for each $G \in \mathcal{G}$ there exists at least two non-empty disjoint proper subsets $V_{1},V_{2}\subseteq V(G)$ satisfying $V(G)\setminus(V_{1} \cup…