For a graph \(G\) of size \(m \geq 1\) and edge-induced subgraphs \(F\) and \(H\) of size \(r\) (\(1 \leq r \leq m\)), the subgraph \(Z\) is said to be obtained from \(F\) by an edge jump if there exist four distinct vertices \(u, v, w\), and \(x\) in \(G\) such that \(uv \in E(F)\), \(wx \in E(G) – E(F)\), and \(H = F – uv + wx\). The minimum number of edge jumps required to transform \(F\) into \(H\) is the jump distance from \(F\) to \(H\). For a graph \(G\) of size \(m \geq 1\) and an integer \(r\) with \(1 \leq r \leq m\), the \(r\)-jump graph \(J_r(G)\) is that graph whose vertices correspond to the edge-induced subgraphs of size \(r\) of \(G\) and where two vertices of \(J_r(G)\) are adjacent if and only if the jump distance between the corresponding subgraphs is \(1\). For \(k \geq 2\), the \(k\)th iterated jump graph \(J^k(G)\) is defined as \(J_r(J^{k-1}_{r}(G))\), where \(J^1_r(G) = J_r(G)\). An infinite sequence \(\{G_i\}\) of graphs is planar if every graph \(G_i\) is planar; while the sequence \(\{G_i\}\) is nonplanar otherwise. It is shown that if \(\{J^k_2(G)\}\) is a nonplanar sequence, then \(J^k_2(G)\) is nonplanar for all \(k \geq 3\) and there is only one graph \(G\) such that \(J^2_2(G)\) is planar. Moreover, for each integer \(r \geq 3\), if \(G\) is a connected graph of size at least \(r + 2\) for which \(\{J^k_r(G)\}\) is a nonplanar sequence, then \(J^k_r(G)\) is nonplanar for all \(k \geq 3\).

Let \(G\) be a finite group written additively and \(S\) a non-empty subset of \(G\). We say that \(S\) is \(e-exhaustive\) if \(G = S + \cdots + S\) (\(e\) times). The minimal integer \(e > 0\), if it exists, such that \(S\) is \(e-exhaustive\), is called the exhaustion number of the set \(S\) and is denoted by \(e(S)\). In this paper, we completely determine the exhaustion numbers of subsets of Abelian groups which are in arithmetic progression. The exhaustion numbers of various subsets of Abelian groups which are not in arithmetic progression are also determined.

Given graphs \(G\) and \(H\), an edge coloring of \(G\) is called an \((H,q)\)-coloring if the edges of every copy of \(H \subset G\) together receive at least \(q\) colors. Let \(r(G,H,q)\) denote the minimum number of colors in a \((H,q)\)-coloring of \(G\). In [6] Erdős and Gyárfás studied \(r(K_n,K_p,q)\) if \(p\) and \(q\) are fixed and \(n\) tends to infinity. They determined for every fixed \(p\) the smallest \(q\) for which \(r(K_n,K_p,q)\) is linear in \(n\) and the smallest \(q\) for which \(r(K_n,K_p,q)\) is quadratic in \(n\). In [9] we studied what happens between the linear and quadratic orders of magnitude. In [2] Axenovich, Füredi, and Mubayi generalized some of the results of [6] to \(r(K_{n,n},K_{p,p},q)\). In this paper, we adapt our results from [9] to the bipartite case, namely we study \(r(K_{n,n},K_p,p,q)\) between the linear and quadratic orders of magnitude. In particular, we show that we can have at most \(\log p + 1\) values of \(q\) which give a linear \(r(K_{n,n},K_{p,p},q)\).

In this paper, we define the concept of generalized Fibonacci polynomial of a graph \(G\) which gives the total number of all \(k\)-stable sets in generalized lexicographical products of graphs. This concept generalizes the Fibonacci polynomial of a graph introduced by G. Hopkins and W. Staton in [3].

A Fibonacci string of order \(n\) is a binary string of length \(n\) with no two consecutive ones. The Fibonacci cube \(\Gamma_n\) is the subgraph of the hypercube \(Q_n\) induced by the set of Fibonacci strings of order \(n\). For positive integers \(i, n\), with \(n \geq i\), the \(i\)th extended Fibonacci cube is the vertex-induced subgraph of \(Q_n\) for which \(V(\Gamma_{i}^{n}) = V_i\) is defined recursively by

\[V_{n+2}^{i} = 0 V_{n+1}^{i} + 10V_n^{i},\]

with initial conditions \(V_i^i = B_i, V_{i+1}^{i} = B_{i+1}\), where \(B_k\) denotes the set of binary strings of length \(k\). In this study, we answer in the affirmative a conjecture of Wu [10] that the sequences \(\{|V_n^i|\}_{i={1+2}}^\infty\) are pairwise disjoint for all \(i \geq 0\), where \(V_n^0 = V(\Gamma_n)\).

Let \(S\) be a simple polygon in the plane whose vertices may be partitioned into sets \(A’, B’\), such that for every two points of \(A’\) (of \(B’\)), the corresponding segment is in \(S\). Then \(S\) is a union of \(6\) (or possibly fewer) convex sets. The number \(6\) is best possible. Moreover, the simple connectedness requirement for set \(S\) cannot be removed.

An \(\lambda\)-design on \(v\) points is a set of \(v\) distinct subsets (blocks) of a \(v\)-element set (points) such that any two different blocks meet in exactly \(\lambda\) points and not all of the blocks have the same size. Ryser’s and Woodall’s \(\lambda\)-design conjecture states that all \(\lambda\)-designs can be obtained from symmetric designs by a certain complementation procedure. The main result of the present paper is that the \(\lambda\)-design conjecture is true when \(v = 8p + 1\), where \(p \equiv 1\) or \(7\) (mod \(8\)) is a prime number.

For an ordered set \(W = \{w_1, w_2, \ldots, w_e\}\) of vertices and a vertex \(v\) in a connected graph \(G\), the representation of \(v\) with respect to \(W\) is the \(e\)-vector \(r(v|W) = (d(v, w_1), d(v, w_2), \ldots, d(v, w_k))\), where \(d(x, y)\) represents the distance between the vertices \(x\) and \(y\). The set \(W\) is a resolving set for \(G\) if distinct vertices of \(G\) have distinct representations with respect to \(W\). A resolving set for \(G\) containing a minimum number of vertices is a basis for \(G\). The dimension \(\dim(G)\) is the number of vertices in a basis for \(G\). A resolving set \(W\) of \(G\) is connected if the subgraph \(\langle W \rangle\) induced by \(W\) is a connected subgraph of \(G\). The minimum cardinality of a connected resolving set in a graph \(G\) is its connected resolving number \(cr(G)\). The relationship between bases and minimum connected resolving sets in a graph is studied. A connected resolving set \(W\) of \(G\) is a minimal connected resolving set if no proper subset of \(W\) is a connected resolving set. The maximum cardinality of a minimal connected resolving set is the upper connected resolving number \(cr^+(G)\). The upper connected resolving numbers of some well-known graphs are determined. We present a characterization of nontrivial connected graphs of order \(n\) with upper connected resolving number \(n-1\). It is shown that for a pair \(a,b\) of integers with \(1 \leq a \leq b\) there exists a connected graph \(G\) with \(cr(G) = a\) and \(cr^+(G) = b\) if and only if \((a,b) \neq (1,4)\) for all \(i > 2\).