Weihua Yang

The toughness of a non-complete graph [Formula: see text], denoted by [Formula: see text], is defined as the maximum real number [Formula: see text] such that [Formula: see text] for every cutset [Formula: see text], where [Formula: see text] denotes the number of components in [Formula: see text]. In this paper, we show that every [Formula: see text]-connected [Formula: see text]-free graph [Formula: see text] with [Formula: see text] is Hamilton-connected. As a direct corollary, every [Formula: see text]-tough [Formula: see text]-free graph is Hamilton-connected. This improves the results of Zheng et al. [Toughness, forbidden subgraphs, and Hamilton-connected graphs. Discuss. Math. Graph Theory, 42(1):187–196, 2021], which states that every [Formula: see text]-free graph [Formula: see text] with [Formula: see text] is Hamilton-connected.

This paper introduces and investigates the connectivity properties with [Formula: see text]-PFIG, a novel extension of fuzzy incidence graphs characterized by multiple membership values corresponding to different polarities. The study systematically defines and explores key connectivity concepts, including paths, cycles, and connectedness, under the [Formula: see text]-polar fuzzy framework. The authors present theoretical findings that expand traditional graph connectivity concepts to a more generalized context, offering a deeper understanding of the behavior and structure of [Formula: see text]-PFIG. This study introduces an innovative method for connectivity concept and incidence graph within [Formula: see text]-polar fuzzy graphs ([Formula: see text]-PFG). Several properties, along with detailed descriptions are thoroughly investigated. The proposed approach accounts for the various merits of each member in a railway network system by considering individual merits (self-weight) in the nature.

Let [Formula: see text] be a graph, and let [Formula: see text] and [Formula: see text] be two integers with [Formula: see text]. A [Formula: see text]-factor of [Formula: see text] is a spanning subgraph in which every connected component is isomorphic to an element of [Formula: see text], where [Formula: see text] is a special family of trees. In this paper, for a graph, we give a sufficient and necessary condition to have [Formula: see text]-factors based on the binding number. In addition, we propose several sufficient conditions for the existence of [Formula: see text]-factors. Furthermore, we show that some bounds are best possible.

For a graph [ ], a function [ ] is a Roman {2}-dominating function (R{2}DF) if [ ] satisfies the following condition: if [ ] then there is [ ] such that [ ] or there is [ ] such that [ ]. Let [ ] denote the set of vertices assigned [ ] by function [ ]. The weight of a R{2}DF [ ] is the sum [ ]. The Roman {2}-domination number is the minimum weight of a R{2}DF on [ ], denoted by [ ]. A function [ ] is an independent Roman {2}-dominating function (IR{2}DF) if [ ] is a R{2}DF and [ ] is an independent set. In this paper, we initially study the influence of edge addition on the Roman {2}-domination number, and for two positive integer [ ] and [ ], we construct a graph [ ] and an induced subgraph [ ] of [ ] such that [ ] and [ ]. Based on this construction, we conclude that there is no relation between the Roman {2}-domination number of a graph and its induced subgraph. Furthermore, we give a bound for the independent Roman [ ]-domination number on proper interval graphs, and a linear-time algorithm for computing the independent Roman [ ]-domination number of unicyclic graphs with restrictions.

A cyclic Gray code is an ordered sequence that contains all binary strings of a given length in which each consecutive string differs from the previous one in exactly one coordinate. Such a sequence directly represents a Hamiltonian cycle and path in the hypercube. Ruskey and Savage asked whether every matching in the hypercube extends to a Hamiltonian cycle, and Fink proved that every perfect matching in [ ] extends to a Hamiltonian cycle. In this study, we address a problem related to Hamiltonian paths. We prove that every matching at most in five directions where every [ ]-cycle contains at least one unsaturated vertex, can be extended to a Hamiltonian path joining two vertices in distinct partite sets of [ ], for [ ]. The only forbidden configurations are the so-called half-layers, which form certain natural obstacles.