In this paper, we consider the relationship between toughness and the existence of \([a, b]\)-factors with inclusion/exclusion properties. We obtain that if \(t(G) \geq a – 1 + \frac{a – 1}{b}\) with \(b > a > 2\), where \(a, b\) are two integers, then for any two given edges \(e_1\) and \(e_2\), there exist an \([a, b]\)-factor including \(e_1, e_2\); and an \([a, b]\)-factor including \(e_1\) and excluding \(e_2\); as well as an \((a, b)\)-factor excluding \(e_1, e_2\). Furthermore, it is shown that the results are best possible in some sense.

In this paper, we will determine the NBB bases with respect to a certain standard ordering of atoms of lattices of \(321\)-\(312\)-\(231\)-avoiding permutations and of \(321\)-avoiding permutations with the weak Bruhat order. Using our expressions of NBB bases, we will calculate the Möbius numbers of these lattices. These values are shown to be related to Fibonacci polynomials.

Let \(D(G)\) denote the signless Dirichlet spectral radius of the graph \(G\) with at least a pendant vertex, and \(\pi_1\) and \(\pi_2\) be two nonincreasing unicyclic graphic degree sequences with the same frequency of number \(1\). In this paper, the signless Dirichlet spectral radius of connected graphs with a given degree sequence is studied. The results are used to prove a majorization theorem of unicyclic graphs. We prove that if \(\pi_1 \unrhd \pi_2\), then \(D(G_1) \leq D(G_2)\) with equality if and only if \(\pi_1 = \pi_2\), where \(G_1\) and \(G_2\) are the graphs with the largest signless Dirichlet spectral radius among all unicyclic graphs with degree sequences \(\pi_1\) and \(\pi_2\), respectively. Moreover, the graphs with the largest signless Dirichlet spectral radius among all unicyclic graphs with \(k\) pendant vertices are characterized.

A function \(f\) is called a graceful labeling of a graph \(G\) with \(m\) edges, if \(f\) is an injective function from \(V(G)\) to \(\{0, 1, 2, \ldots, m\}\) such that when every edge \(uv\) is assigned the edge label \(|f(u) – f(v)|\), then the resulting edge labels are distinct. A graph which admits a graceful labeling is called a graceful graph. A graceful labeling of a graph \(G\) with \(m\) edges is called an \(\alpha\)-labeling if there exists a number \(\alpha\) such that for any edge \(uv\), \(\min\{f(u), f(v)\} \leq \lambda < \max\{f(u), f(v)\}\). The characterization of graceful graphs appears to be a very difficult problem in Graph Theory. In this paper, we prove a basic structural property of graceful graphs, that every tree is a subtree of a graceful graph, an \(\alpha\)-labeled graph, and a graceful tree, and we discuss a related open problem towards settling the popular Graceful Tree Conjecture.

We use rook placements to prove Spivey’s Bell number formula and other identities related to it, in particular, some convolution identities involving Stirling numbers and relations involving Bell numbers. To cover as many special cases as possible, we work on the generalized Stirling numbers that arise from the rook model of Goldman and Haglund. An alternative combinatorial interpretation for the Type II generalized \(q\)-Stirling numbers of Remmel and Wachs is also introduced, in which the method used to obtain the earlier identities can be adapted easily.

An \(H\)-triangle is a triangle with corners in the set of vertices of a tiling of \(\mathbb{R}^2\) by regular hexagons of unit edge. Let \(b(\Delta)\) be the number of the boundary \(H\)-points of an \(H\)-triangle \(\Delta\). In [3] we made a conjecture that for any \(H\)-triangle with \(k\) interior \(H\)-points, we have \(b(\Delta) \in \{3, 4, \ldots, 3k+4, 3k+5, 3k+7\}\). In this note, we prove the conjecture is true for \(k = 4\), but not true for \(k = 5\) because \(b(\Delta)\) cannot equal \(15\).