A tree could be defined as follows. An edge is a tree. If \(T_{k-1}=\cup_{i=1}^{k-1}e_i\) is a tree with \(k-1\) edges \(e_i\), and \(e_k\) an edge, then \(T_k=T_{k-1}\cup e_k\) is a tree if \(T_{k-1}\cap e_k\) is a point. We generalize this construction: A simplex \(S_1\) of dimension \(\ge1\) is a thick tree. If \(G_{k-1}=\cup_{i=1}^{k-1}S_i\) is a thick tree, where \(S_i\) are simplices of dimension \(\ge1\), and \(S_k\) a new simplex of dimension \(\ge1\), then \(G_{k-1}\cup S_k\) is a thick tree if \(G_{k-1}\cap S_k\) is a point. All homological properties of Stanley-Reisner rings of thick trees are well known. We determine the Hilbert series and Betti numbers for Stanley-Reisner rings of skeletons of thick trees. From this one can read of projective dimension, regularity, and judge when they are Cohen-Macaulay.

In this paper we introduce and study the hyper-Mersenne numbers, a class of integer sequences extending the classical Mersenne numbers which arise in a combinatorial and algebraic context. Using generating functions, we derive explicit formulae and identities for these sequences. In particular, we find relations to binomial coefficients and figurate numbers. We also provide a closed-form expression for the determinant of associated matrices, valid in full generality.

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.