A bipartite labeling of a tree of order \(n\) is a bijective function that identifies the vertices of \(T\) with the elements of \(\{0, 1, \dots, n-1\}\) in such a way that there exists an integer \(\lambda\) such that the set of labels on the stable sets of \(T\) are \(\{0,1, \dots, \lambda\}\) and {\(\lambda + 1, \lambda +2. \dots, n-1\}.\) The most restrictive and versatile bipartite labeling is the variety called \(\alpha\text{-labeling}\). In this work we present a new construction of \(\alpha\text{-labeled}\) trees where any two adjacent vertices of a path-like tree, or a similar caterpillar, can be amalgamated with selected vertices of two equivalent trees.

A set \(D\) of vertices of a graph \(G=(V_G,E_G)\) is a dominating set of \(G\) if every vertex in \(V_G-D\) is adjacent to at least one vertex in \(D\). The domination number of a graph \(G\), denoted by \(\gamma(G)\), is the cardinality of a smallest dominating set of \(G\). A subset \(D\subseteq V_G\) is called a certified dominating set of \(G\) if \(D\) is a dominating set of \(G\), and every vertex in \(D\) has either zero or at least two neighbours in \(V_G-D\). The cardinality of a smallest certified dominating set of \(G\) is called the certified domination number of \(G\), and it is denoted by \(\gamma_{\rm cer}(G)\). A vertex \(v\) of \(G\) is certified critical if \(\gamma_{\rm cer}(G -v) < \gamma_{\rm cer}(G)\), and a graph \(G\) is vertex certified domination critical or \(\gamma_{cer}\)–critical if the removal of any vertex reduces its certified domination number. In this paper, we give examples and properties of certified critical vertices and vertex certified domination critical graphs. As an example of an application of certified critical vertices, we give a constructive characterisation of trees for which the smaller partite set is a minimum certified dominating set.

In this paper, we consider Ramsey and Gallai-Ramsey numbers for a generalized fan \(F_{t,n}:=K_1+nK_t\) versus triangles. Besides providing some general lower bounds, our main results include the evaluations of \(r(F_{3,2}, K_3)=13\) and \(gr(F_{3,2}, K_3, K_3)=31\).

Seymour’s Second Neighborhood Conjecture (SSNC) asserts that every finite oriented graph has a vertex \(v\) whose second out-neighborhood is at least as large as its first out-neighborhood. Such a vertex is called a Seymour vertex. In this note, we introduce pseudo-Seymour set such that Seymour’s conjecture becomes: Every oriented graph has a singleton pseudo-Seymour set. We prove that any oriented graph has a pseudo-Seymour set \(S\) with \(|S|=2\). Furthermore, we show that there are pseudo-Seymour sets of any size at least 2. We define \(\rho\)-Seymour vertex where \(0 < \rho \leq 1\), and give an approach such that finding \(\rho=1\) is equivalent to the existence of Seymour vertex. Attempting to maximize its value, we prove that for any oriented graph with minimum out-degree \(\delta\), there is \(\rho=\frac{2}{3}(1+\frac{1}{2\delta})\).

In this paper, given a homeomorphism \(f\) of a compact metric space \(X\), we show that the set of all asymptotic average shadowable points of \(f\) is an open and invariant set and \(f\) has the asymptotic average shadowing property if and only if the set of all asymptotic average shadowable points of \(f\) is \(X\) if and only if any Borel probability measure \(\mu\) of \(X\) has the asymptotic average shadowing property.

This paper is concerned with the pullback attractors for the Kirchhoff type BBM equations defined on unbounded domains. Sobolev embeddings are invalid on unbounded domains. We obtain the pullback asymptotic compactness of such non-autonomous BBM equations by using the method of uniform tail-estimates.

We say that a graph \(G\) has a path-system with respect to a set \(W\) of even number of vertices in \(G\) if \(G\) has vertex-disjoint paths \(P_1,P_2, \ldots, P_m\) such that (i) each path \(P_i\) connects two vertices of \(W\) and (ii) the set of end-vertices of the paths \(P_i\) is exactly \(W\). In particular, \(m=|W|/2\). Moreover, if \(G\) has a path-system with respect to every set \(W\) of even number of vertices in \(G\), we say that \(G\) has a path system. We prove the following theorems: (i) if \(G\) is an \(r\)-edge-connected \(r\)-regular graph, then for any \(r-1\) edges \(e_1,\ldots, e_{r-1}\), \(G-\{e_1,\ldots, e_{r-1}\}\) has a path-system, (ii) every \(k\)-connected \(K_{1,k+1}\)-free graph has a path-system, and (iii) if a connected bipartite graph \(G\) with bipartition \((A,B)\) satisfies \(|A| \le 2|B|\), \(|N_G(X)| \ge 2|X|\) or \(N_G(X)=B\) for all \(X\subseteq A\), and \(|N_G(Y)| \ge |Y|\) or \(N_G(Y)=A\) for all \(Y\subseteq B\), then \(G\) has a path-system with respect to every set \(W\) of even number of vertices of \(A\).

For a graph \(G\), an Italian dominating function (IDF) is a function \(f: V(G) \rightarrow \{0,1,2\}\) such that all vertices labeled with 0 must have at least two neighbors assigned the label 1 or at least one neighbor assigned the label 2. The weight of \(f\), denoted by \(w(f)\), is calculated by summing all the labels assigned by the function. Let \(f\) be an IDF on \(G\) with a minimum weight, denoted as \(\gamma_I(G)\). If \(S\) is the set of vertices where \(f(v) > 0\), then an Inverse Italian Dominating Function (IIDF) \(f'\) is defined as an IDF on \(G\) such that \(f'(v) = 0\) for all \(v \in S\). The notation \(\gamma_{iI}(G)\) represents the Inverse Italian Domination Number of the graph \(G\), which is the minimum weight among all IIDFs on \(G\). In this paper, we find \(\gamma_{iI}(G)\) of graphs and characterize the graphs for which \(\gamma_I(G) = 2\) and \(3\), as well as those with \(\gamma_{iI}(G) = 2\) and \(3\). Additionally, we provide a characterization of trees and graphs that achieve the largest possible \(\gamma_{iI}(G)\).