Let \(G = (V,E)\) be an n-vertex graph and \(f : V \rightarrow \{1,2,\ldots,n\}\) be a bijection. The additive bandwidth of \(G\), denoted \(B^+(G)\), is given by \(B^+(G) = \min_{f} \max_{u,v\in E} |f(u) + f(v) – (n+1)|\), where the minimum ranges over all possible bijections \(f\). The additive bandwidth cannot decrease when an edge is added, but it can increase to a value which is as much as three times the original additive bandwidth. The actual increase depends on \(B^+(G)\) and n and is completely determined.

In Minimal Enclosings of Triple Systems I, we solved the problem of minimal enclosings of \(\text{BIBD}(v, 3, \lambda)\) into \(\text{BIBD}(v+1, 3, \lambda+m)\) for \(1 \leq \lambda \leq 6\) with a minimal \(m \geq 1\). Here we consider a new problem relating to the existence of enclosings for triple systems for any \(v\), with \(1 < 4 < 6\), of \(\text{BIBD}(v, 3, \lambda)\) into \(\text{BIBD}(v+s, 3, \lambda+1)\) for minimal positive \(s\). The non-existence of enclosings for otherwise suitable parameters is proved, and for the first time the difficult cases for even \(\lambda\) are considered. We completely solve the case for \(\lambda \leq 3\) and \(\lambda = 5\), and partially complete the cases \(\lambda = 4\) and \(\lambda = 6\). In some cases a \(1\)-factorization of a complete graph or complete \(n\)-partite graph is used to obtain the minimal enclosing. A list of open cases for \(\lambda = 4\) and \(\lambda = 6\) is attached.

Halin’s Theorem characterizes those locally finite infinite graphs that embed in the plane without accumulation points by giving a set of six topologically-excluded subgraphs. We prove the analogous theorem for graphs that embed in an open Möbius strip without accumulation points. There are \(153\) such obstructions under the ray ordering defined herein. There are \(350\) obstructions under the minor ordering. There are \(1225\) obstructions under the topological ordering. The relationship between these graphs and the obstructions to embedding in the projective plane is similar to the relationship between Halin’s graphs and \(\{K_5, K_{3,3}\}.^1\)

In [5] Pila presented best possible sufficient conditions for a regular \(\sigma\)-connected graph to have a \(1\)-factor, extending a result of Wallis [7]. Here we present best possible sufficient conditions for a \(\sigma\)-connected regular graph to have a \(k\)-factor for any \(k \geq 2\).

We find a maximal number of directed circuits (directed cocircuits) in a base of a cycle (cut) space of a digraph. We show that this space has a base composed of directed circuits (directed cocircuits) if and only if the digraph is totally cyclic (acyclic). Furthermore, this basis can be considered as an ordered set so that each element of the basis has an arc not contained in the previous elements.

In this paper, we show that if \(G\) is a harmonious graph, then \((2n+1)G\) (the disjoint union of \(2n+1\) copies of \(G\)) and \(G ^{(2n+1)}\) (the graph consisting of \(2n+1\) copies of \(G\) with one fixed vertex in common) are harmonious for all \(n \geq 0\).

A critical set in a Latin square of order \(n\) is a set of entries from the square which can be embedded in precisely one Latin square of order \(n\), such that if any element of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order \(n\). In this paper we find all the critical sets of different sizes in the Latin squares of order at most six. We count the number of main and isotopy classes of these critical sets and classify critical sets from the main classes into various “strengths”. Some observations are made about the relationship between the numbers of classes, particularly in the \(6 \times 6\) case. Finally some examples are given of each type of critical set.

A proper vertex \(k\)-coloring of a graph \(G\) is dynamic if for every vertex \(v\) with degree at least \(2\), the neighbors of \(v\) receive at least two different colors. The smallest integer \(k\) such that \(G\) has a dynamic \(k\)-coloring is the dynamic chromatic number \(\chi_d(G)\). We prove in this paper the following best possible upper bounds as an analogue to Brook’s Theorem, together with the determination of chromatic numbers for complete \(k\)-partite graphs.

1. If \(\Delta \leq 3\), then \(\chi_d(G) \leq 4\), with the only exception that \(G = C_5\), in which case \(\chi_d(C_5) = 5\).
2. If \(\Delta \geq 4\), then \(\chi_d(G) \leq \Delta + 1\).
3. \(\chi_d(K_{1,1}) = 2\), \(\chi_d(K_{1,m}) = 3\) and \(\chi_d(K_{m,n}) = 4\) for \(m, n \geq 2\); \(\chi_d(K_{k,n_1,n_2,\ldots,n_k}) = k\) for \(k \geq 3\).

If \(x\) is a vertex of a digraph \(D\), then we denote by \(d^+(x)\) and \(d^-(x)\) the outdegree and the indegree of \(x\), respectively. The global irregularity of a digraph \(D\) is defined by \(i_g(D) = \max\{d^+(x),d^-(x)\} – \min\{d^+(y),d^-(y)\}\) over all vertices \(x\) and \(y\) of \(D\) (including \(x = y\)). If \(i_g(D) = 0\), then \(D\) is regular and if \(i_g(D) \leq 1\), then \(D\) is almost regular.

A \(c\)-partite tournament is an orientation of a complete \(c\)-partite graph. It is easy to see that there exist regular \(c\)-partite tournaments with arbitrarily large \(c\) which contain arcs that do not belong to a directed cycle of length \(3\). In this paper we show, however, that every arc of an almost regular \(c\)-partite tournament is contained in a directed cycle of length four, when \(c \geq 8\). Examples show that the condition \(c \geq 8\) is best possible.

We address the following problem: What minimum degree forces a graph on \(n\) vertices to have a cycle with at least \(c\) chords? We prove that any graph with minimum degree \(\delta\) has a cycle with at least \(\frac{(\delta+1)(\delta-2)}{2}\) chords. We investigate asymptotic behaviour for large \(n\) and \(c\) and we consider the special case where \(n = c\).

We prove that a finite set \(A\) of points in the \(n\)-dimensional Euclidean space \(\mathcal{R}^n\) is uniquely determined up to translation by three of its subsets of cardinality \(|A|-1\) given up to translation, i.e. the Reconstruction Number of such objects is three. This result is best-possible.

We solve the problem of existence of minimal enclosings for triple systems with \(1 \leq \lambda \leq 6\) and any \(v\), i.e., an inclusion of \(\text{BIBD}(v, 3, \lambda)\) into \(\text{BIBD}(v+1, 3, \lambda+m)\) for minimal positive \(m\). A new necessary general condition is derived and some general results are obtained for larger \(\lambda\) values.

Colour the edges of a \(K_{24n+1}\) by \(12\) colours so that every vertex in every colour has degree \(2n\). Is there a totally multicoloured \(C_4\) (i.e. every edge gets a different colour)? Here we answer in the affirmative to this question. In [1] P. Erdős stated the same problem for \(K_{12n+1}\) and \(6\) colours, it was settled in [2].

In this paper we follow the terminology and symbols of [3]. We assume the complete graph \(K_{24n+1}\) to have the vertex-set \(V=V(K_{24n+1}) = \{1, 2, \ldots, 24n+1\}\).

Let \(P(G)\) denote the chromatic polynomial of a graph \(G\). Two graphs \(G\) and \(H\) are chromatically equivalent, written \(G \sim H\), if \(P(G) = P(H)\). A graph \(G\) is chromatically unique if for any graph \(H\), \(G \sim H\) implies that \(G\) is isomorphic with \(H\). In “Chromatic Equivalence Classes of Certain Generalized Polygon Trees”, Discrete Mathematics Vol. \(172, 108–114 (1997)\), Peng \(et\; al\). studied the chromaticity of certain generalized polygon trees. In this paper, we present a chromaticity characterization of another big family of such graphs.

The step domination number of all graphs of diameter two is determined.

We use generator matrices \(G\) satisfying \(GG^T = aI + bJ\) over \(\mathbb{Z}_k\) to obtain linear self-orthogonal and self-dual codes. We give a new family of linear self-orthogonal codes over \(\text{GF}(3)\) and \(\mathbb{Z}_4\) and a new family of linear self-dual codes over \(\text{GF}(3)\).

Let \(S\) be a simply connected orthogonal polygon in the plane. Assume that the vertex set of \(S\) may be partitioned into sets \(A, B\) such that for every pair \(x, y\) in \(A\) (in \(B\)), \(S\) contains a staircase path from \(x\) to \(y\). Then \(S\) is a union of two or three orthogonally convex sets. If \(S\) is star-shaped via staircase paths, the number two is best, while the number three is best otherwise. Moreover, the simple connectedness requirement cannot be removed. An example shows that the segment visibility analogue of this result is false.

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 \(i > 2\).