Some designs using the action of the linear fractional groups \(L_2(q)\), \(q = 11, 13, 16, 17, 19, 23\) are constructed. We will show that \(L_2(q)\) or its automorphism group acts as the full automorphism group of each of the constructed designs except in the case \(q = 16\). For designs constructed from \(L_2(16)\), we will show that \(L_2(16)\), \(L_2(16) : 2\), \(L_2(16) : 4\) or \(S_{17}\) can arise as the full automorphism group of the design.

For odd \(n \geq 5\), the Flower Snark \(F_n = (V, E)\) is a simple undirected cubic graph with \(4n\) vertices, where \(V = \{a_i : 0 \leq i \leq n-1\} \cup \{b_i : 0 \leq i \leq n-1\} \cup \{c_i : 0 \leq i \leq 2n-1\}\) and \(E = \{b_ib_{(i+1)\mod(n)}: 0 \leq i \leq n-1\} \cup \{c_ic_{(i+1)\mod(2n)} : 0 \leq i \leq 2n-1\} \cup \{a_ib_i,a_ic_i,a_ic_{n+i} : 0 \leq i \leq n-1\}\). For \(n = 3\) or even \(n \geq 4\), \(F_n\) is called the related graph of Flower Snark. We show that the crossing number of \(F_n\) equals \(n – 2\) if \(3 \leq n \leq 5\), and \(n\) if \(n \geq 6\).

A subset \(S\) of the vertex set of a graph \(G\) is called acyclic if the subgraph it induces in \(G\) contains no cycles. We call \(S\) an acyclic dominating set if it is both acyclic and dominating. The minimum cardinality of an acyclic dominating set, denoted by \(\gamma_a(G)\), is called the acyclic domination number of \(G\). A graph \(G\) is \({2-diameter-critical}\) if it has diameter \(2\) and the deletion of any edge increases its diameter. In this paper, we show that for any positive integers \(k\) and \(d \geq 3\), there is a \(2\)-diameter-critical graph \(G\) such that \(\delta(G) = d\) and \(\gamma_a(G) – \delta(G) \geq k\), and our result answers a question posed by Cheng et al. in negative.

A function \(f: V \to \{1,\ldots,k\}\) is a broadcast coloring of order \(k\) if \(\pi(u) = \pi(v)\) implies that the distance between \(u\) and \(v\) is more than \(\pi(u)\). The minimum order of a broadcast coloring is called the broadcast chromatic number of \(G\), and is denoted \(\chi_b(G)\). In this paper we introduce this coloring and study its properties. In particular, we explore the relationship with the vertex cover and chromatic numbers. While there is a polynomial-time algorithm to determine whether \(\chi_b(G) \leq 3\), we show that it is \(NP\)-hard to determine if \(\chi_b(G) \leq 4\). We also determine the maximum broadcast chromatic number of a tree, and show that the broadcast chromatic number of the infinite grid is finite.

A connected graph \(G = (V, E)\) is said to be \((a,d)\)-antimagic if there exist positive integers \(a,d\) and a bijection \(f : E \to \{1,2,\ldots,|E|\}\) such that the induced mapping \(g_f : V \to \mathbb{N}\), defined by \(g_f(v) = \sum f(uv)\),\({uv \in E(G)}\) is injective and \(g_f(V) = \{a,a+d,\ldots,a+(|V|-1)d\}\). Mirka Miller and Martin Bača proved that the generalized Petersen graph \(P(n, 2)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for \(n \equiv 0 \pmod{4}\), \(n \geq 8\) and conjectured that the generalized Petersen graph \(P(n, k)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for even \(n\) and \(2 \leq k \leq \frac{n}{2}-1\). In this paper, we show that the generalized Petersen graph \(P(n, 3)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for even \(n \geq 8\).

In this paper, we derive new recurrence relations and generating matrices for the sums of usual Tribonacci numbers and \(4n\) subscripted Tribonacci sequences, \(\{T_{4n}\}\), and their sums. We obtain explicit formulas and combinatorial representations for the sums of terms of these sequences. Finally, we represent relationships between these sequences and permanents of certain matrices.

Let \(\mathcal{K} = (K_{ij})\) be an infinite lower triangular matrix of non-negative integers such that \(K_{i0} = 1\) and \(K_{ii} \geq 1\) for \(i \geq 0\). Define a sequence \(\{V_i(\mathcal{K})\}_{m\geq0}\) by the recurrence \(V_{i+1}(\mathcal{K}) = \sum_{j=0}^m K_{mj}V_j(\mathcal{K})\) with \(V_0(\mathcal{K}) = 1\). Let \(P(n;\mathcal{K})\) be the number of partitions of \(n\) of the form \(n = p_1 + p_2 + p_3 + p_4 + \cdots\) such that \(p_j \geq \sum_{i\geq j} K_{ij}p_{i+1}\) for \(j \geq 1\) and let \(P(n;V(\mathcal{K}))\) denote the number of partitions of \(n\) into summands in the set \(V(\mathcal{K}) = \{V_1(\mathcal{K}), V_2(\mathcal{K}), \ldots\}\). Based on the technique of MacMahon’s partitions analysis, we prove that \(P(n;\mathcal{K}) = P(n;V(\mathcal{K}))\) which generalizes a recent result of Sellers’. We also give several applications of this result to many classical sequences such as Bell numbers, Fibonacci numbers, Lucas numbers, and Pell numbers.

Minimal blocking sets of class \([h,k]\) with respect to the external lines to an elliptic quadric of \(\text{PG}(3,q)\), \(q \geq 5\) prime, are characterized.

For every integer \(c\) and every positive integer \(k\), let \(n = r(c, k)\) be the least integer, provided that it exists, such that for every coloring

\[\Delta: \{1,2,\ldots,n\} \rightarrow \{0,1\},\]

there exist three integers, \(x_1, x_2, x_3\), (not necessarily distinct) such that
\[\Delta(x_1) = \Delta(x_2) = \Delta(x_3)\]
and
\[x_1+x_2+c= kx_3.\]

If such an integer does not exist, then let \(r(c, k) = \infty\). The main result of this paper is that

\[r(c,2) =
\begin{cases}
|c|+1 & \text{if } c \text{ is even} \\
\infty & \text{if } c \text{ is odd}
\end{cases}\]

for every integer \(c\). In addition, a lower bound is found for \(r(c, k)\) for all integers \(c\) and positive integers \(k\) and linear upper and lower bounds are found for \(r(c, 3)\) for all positive integers \(c\).

Let \(C_n\) denote the cycle with \(n\) vertices, and \(C_n^{(t)}\) denote the graphs consisting of \(t\) copies of \(C_n\) with a vertex in common. Koh et al. conjectured that \(C_n^{(t)}\) is graceful if and only if \(nt \equiv 0,3 \pmod 4\). The conjecture has been shown true for \(n = 3,5,6,7,4k\). In this paper, the conjecture is shown to be true for \(n = 9\).