For positive integers \(c \geq 0\) and \(k \geq 1\), let \(n = R(c, k)\) be the least integer, provided it exists, such that every \(2\)-coloring of the set \([1,n] = \{1,\ldots,n\}\) admits a monochromatic solution to the equation \(x + y+c = 4z\) with \(x, y, z \in [1,n]\). In this paper, the precise value of \(R(c, 4)\) is shown to be \(\left\lceil{3c + 2}/{8}\right\rceil\) for all even \(c \geq 34\).

Given a positive integer \(n\) such that \(-1\) is a quadratic residue mod \(n\), we give an algorithm that computes the integers \(u\) and \(v\) which satisfy the equation \(n = u^2 + v^2\). To do this, we use the group structure of the Modular group \(\Gamma= \text{PSL}(2,\mathbb{Z})\).

For a graph \(G = (V(G),E(G))\), a set \(S \subseteq V(G)\) is called a dominating set if \(N_G[S] = V(G)\). A dominating set \(S\) is said to be minimal if no proper subset \(S’ \subset S\) is a dominating set. Let \(\gamma(G)\) (called the domination number) and \(\Gamma(G)\) (called the upper domination number) be the minimum cardinality and the maximum cardinality of a minimal dominating set of \(G\), respectively. For a tree \(T\) of order \(n \geq 2\), it is obvious that \(1 = \gamma(K_{1,n-1}) \leq \gamma(T) \leq \Gamma(T) \leq \Gamma(K_{1,n-1}) = n-1\). Let \(t(n) = \min_{|T|=n}(\Gamma(T)-\gamma(T))\). In this paper, we determine \(t(n)\) for all natural numbers \(n\). We also characterize trees \(T\) with \(\Gamma(T) – \gamma(T) = t(n)\).

The signless \(r\)-associated Stirling numbers of the first kind \(d_r(n, k)\) counts the number of permutations of the set \(\{1,2,\ldots,n\}\) that have exactly \(k\) cycles, each of which is of length greater than or equal to \(r\), where \(r\)is a fixed positive integer. F. Brenti obtained that the generating polynomials of the numbers \(d_r(n, k)\) have only real zeros. Here we consider the location of zeros of these polynomials.

A kite-design of order \(n\) is a decomposition of the complete graph \(K_n\) into kites. Such systems exist precisely when \(n \equiv 0,1 \pmod{8}\). Two kite systems \((X,\mathcal{K}_1)\) and \((X,\mathcal{K}_2)\) are said to intersect in \(m\) pairwise disjoint blocks if \(|\mathcal{K}_1 \cap \mathcal{K}_2| = m\) and all blocks in \(\mathcal{K}_1 \cap \mathcal{K}_2\) are pairwise disjoint. In this paper, we determine all the possible values of \(m\) such that there are two kite-designs of order \(n\) intersecting in \(m\) pairwise disjoint blocks, for all \(n \equiv 0,1 \pmod{8}\).

In this note, we present some upper bounds for the \(k\)th largest eigenvalue of the adjacency matrix as well as the Laplacian matrix of graphs. Special attention is paid to the Laplacian matrix of trees.

Let \(P(G, \lambda)\) denote the chromatic polynomial of a graph \(G\). Two graphs \(G\) and \(H\) are chromatically equivalent, written \(G \sim H\), if \(P(G, \lambda) = P(H, \lambda)\). A graph \(G\) is chromatically unique, written \(x\)-unique, if for any graph \(H\), \(G \sim H\) implies that \(G\) is isomorphic with \(H\). In this paper, we prove that the graph \(\theta(a_1, a_2, \ldots, a_6)\) is \(x\)-unique for exactly two distinct values of \(a_1, a_2, \ldots, a_6\).

In this paper, we give an explicit expression of the genus distributions of \(M_j^n\), for \(j = 1, 2, \ldots, 11\), which are introduced in the previous paper “Orientable embedding distributions by genus for certain types of non-planar graphs”. For a connected graph \(G = (V, E)\) with a cycle, let \(e\) be an edge on a cycle. By adding \(2n\) vertices \(u_1, u_2,u_3 \ldots, u_n, v_1, v_2,v_3 \ldots, v_n\) on \(e\) in sequence and connecting \(u_k, v_k\) for \(1 \leq k \leq n\), a non-planar graph \(G_n\) is obtained for \(n \geq 3\). Thus, the orientable embedding distribution of \(G_n\) by genus is obtained via the genus distributions of \(M_j^n\).

A graph \(G\) is \(N^m\)-locally connected if for every vertex \(v\) in \(G\), the vertices not equal to \(v\) and with distance at most \(m\) to \(v\) induce a connected subgraph in \(G\). In this note, we first present a counterexample to the conjecture that every \(3\)-connected, \(N^2\)-locally connected claw-free graph is hamiltonian and then show that both connected \(N^2\)-locally connected claw-free graph and connected \(N^3\)-locally connected claw-free graph with minimum degree at least three have connected even \([2, 4]\)-factors.

In J.-P. Serre’s \(Lettre \;à\; M. Tsfasman\) \([3]\), an interesting bound for the maximal number of points on a hypersurface of the \(n\)-dimensional projective space \(PG(n,q)\) over the Galois field \(GF(q)\) with \(q\) elements is given. Using essentially the same combinatorial technique as in \([3]\), we provide a bound which is relative to the maximal dimension of a subspace of \(PG(n,q)\) which is completely contained in the hypersurface. The lower that dimension, the better the bound. Next, by using a different argument, we derive a bound which is again relative to the maximal dimension of a subspace of \(PG(n, q)\) which is completely contained in the hypersurface, If that dimension increases for the latter case, the bound gets better.
As such, the bounds are complementary.