A graph \(G\) is \(H\)-saturated if \(G\) does not have \(H\) as a subgraph but \(G + uv\) has at least one copy of \(H\) for any edge \(uv \notin E(G)\). The smallest number of edges of all \(H\)-saturated graphs of order \(n\) is called \(H\)-saturation number and is denoted by \(sat(n; H)\). In this paper, we establish the existence of \(C_{4}\)-saturated graphs with prescribed the number of edges in some length that is close to \(sat(n; C_{4})\).

The Hall number \(h(G)\) of a graph \(G\) is the minimum integer \(k\) such that every \(k\)-list assignment satisfying Hall’s condition on all induced subgraphs of \(G\) admits a proper coloring. In this paper, we investigate graphs for which the Hall number strictly captures list colorability, satisfying the equality \(h(G)=ch(G)\). We confirm a conjecture of Allagan by proving that this equality holds for every complete multipartite graph without singleton parts. For complete \(k\)-partite graphs of the form \(K(m,n,1,\dots,1)\), we establish that \(h(G)=ch(G)\) for all sufficiently large \(n\). Furthermore, we also determine \(h(G)\) for \(2\)-trees and wheel graphs \(W_n\). We show that for a \(2\)-tree \(G\), \(h(G) \in \{1, 2, 3\}\) for \(|V(G)| = 3, 4\), and \(\ge 5\), respectively. For wheel graphs, we demonstrate that \(h(W_n)\) is dictated by the parity of the rim: \(h(W_n)=3\) for odd \(n\ge5\), and \(h(W_n)=4\) for even \(n\ge6\).

For a connected graph \(G\) of order at least two, a total monophonic set of a graph \(G\) is a monophonic set \(S\) such that the subgraph \(G[S]\) induced by \(S\) has no isolated vertices. The minimum cardinality of a total monophonic set of \(G\) is the total monophonic number of \(G\) and is denoted by \(m_{t}(G)\). We determine bounds for it and characterize graphs which realize the lower bound. Also, some general properties satisfied by this concept are studied. It is shown that for positive integers \(a, b\) such that \(3 \leq a \leq b\) with \(b \leq 2a\), there exists a connected graph \(G\) such that \(m(G) = a\) and \(m_t(G) = b\). Further, if \(p, a, b\) are positive integers such that \(4 \leq a \leq b \leq p\), then there exists a connected graph \(G\) of order \(p\) with \(m_t(G) = a\) and \(m_c(G) = b\), where \(m_c(G)\) is the connected monophonic number of \(G\).

The concept of k-extendability is a fundamental notion in matching theory and is closely related to factor-criticality. In a seminal work, Zhang et al. established sharp conditions under which these two concepts are equivalent. In this paper, we study the equivalence between extendability and factor-criticality from the perspective of Sachs subgraphs and discuss conditions under which these notions are equivalent.

A graph \(G=(V,E)\) is said to be an absolute mean graceful graph if there exists a one-to-one function \(f:V(G)\to \{0,\pm1,\pm2,\ldots,\pm|E(G)|\}\) such that the induced edge-labeling function \(f^*:E(G)\to \{1,2,\ldots,|E(G)|\}\), defined by \[f^*(xy)=\left\lceil{\dfrac{|f(x)-f(y)|}{2}}\right\rceil,\] is bijective. The labeling function \(f\) is called an absolute mean graceful labeling of the graph \(G\). In this paper, we obtain absolute mean graceful labelings for \(m\)-splitting and \(m\)-shadow graphs of various graphs.

There are a number of variations of proper edge-colourings of graphs with restrictions on the subgraphs induced by two colour classes. Deciding whether a graph has a 3-edge-colouring with no 2-coloured 4-cycle is NP-complete for cubic graphs with chromatic index 3. But for bipartite cubic graphs the problem is solved completely in this paper. Furthermore, the minimum number of 2-coloured 4-cycles in a 3-edge-colouring is determined for any subcubic bipartite multigraph.

Let \(F_n:=K_1+nK_2\) be a fan of order \(2n+1\). For \(1\le s<t\), we consider the weakened Gallai-Ramsey number \(gr^t_s(F_n)\), defined to be the least \(p\in \mathbb{N}\) such that every Gallai \(t\)-coloring of \(K_p\) contains a subgraph isomorphic to \(F_n\) whose edges use at most \(s\) colors. Our main results include the evaluations \(gr^t_2(F_2)=t+3\), \(gr^3_2(F_3)=9\), and \(gr^t_{2n-1}(F_n)=2n+1\).

We introduce a parity–sum statistic on permutations that assigns to each position a weight determined by the parity of the entry occupying it. When restricted to alternating permutations, this statistic yields two \(q\)–analogues of the Euler numbers, corresponding to the up–down and down–up types. We establish symmetry and reciprocity properties of these polynomials. Specializing at \(q=1\), the resulting recursions reduce to the classical enumerative relations and recover Andr’e’s convolution identity for the Euler numbers. The distribution of the parity–sum statistic over the full symmetric group is also determined.

Previous work by Lesniak (1975) and recently by Qiao and Zhan (2022) established a sharp lower bound for the number of leaves of a tree as a function of its order and diameter. In this note, we derive a lower bound for the number of leaves as a function of the entire sequence of eccentricities, and provide a complete characterisation of all trees attaining equality. We also obtain a new but simpler proof for the diameter-bound. Furthermore, we establish the analogous result for the maximum with respect to the entire eccentric sequence.

Soft set theory, first introduced by Molodtsov, is a flexible approach for handling uncertainty-related problems and modeling uncertain information. Since soft set operations form the core of the theory, their algebraic properties and related structures have attracted considerable research interest. Several forms of symmetric difference operations have been proposed, including restricted and extended symmetric difference operations. Although restricted symmetric difference has already been defined, its definition is not fully consistent with the general formula of restricted soft set operations. In this paper, we first provide an alternative definition of restricted symmetric difference that follows the general form of restricted soft set operations. We then investigate its algebraic properties together with the extended symmetric difference operation, both for soft sets with a fixed parameter set and for soft sets over \(U\). We also establish new properties analogous to the symmetric difference operation in classical set theory, including the case where parameter sets may be disjoint. By deriving distributive rules, we show that important algebraic structures arise when restricted or extended symmetric difference operations are combined with other soft set operations. This study fills a gap in the literature, guides readers new to the theory, and presents a comprehensive analysis of restricted and extended symmetric difference operations, including corrected theorems and proofs from earlier studies.