Many different approaches exist in studying graphs with high connectivity and small diameter. We consider the effect of deleting vertices and edges from a graph while maintaining a small diameter. The following property is introduced: A graph \( G \) has property \( B_{d,i,j} \) if and only if after the removal of at most \( i \) vertices and at most \( j \) edges, the resulting graph has diameter at most \( d \) and is not the trivial graph on one vertex. The central theme of this paper is to investigate the structure of graphs that have property \( B_{d,i,j} \) and to investigate the structure that is needed to imply that a graph has property \( B_{d,i,j} \). Lower bounds on minimum degree and connectivity that imply property \( B_{d,i,j} \) for specific values of \( d \) are found. These bounds are also shown to be sharp in all but one case.

An \( m \)-cycle system of order \( v \), denoted by \( mCS(v) \), is a decomposition of the complete graph \( K_v \) into \( m \)-cycles. We discuss two types of large sets of \( mCS(v) \) and construct examples of both types for \( (m,v) = (4,9) \) and one type for \( (m,v) = (6,9) \). These are the first large sets of cycle systems constructed with \( m > 3 \), apart from the Hamiltonian cycle decompositions given in [2].

For two vertices \( u \) and \( v \) in a connected graph \( G \), the detour distance \( D(u,v) \) from \( u \) to \( v \) is defined as the length of a longest \( u-v \) path in \( G \). The detour eccentricity \( e_D(v) \) of a vertex \( v \) in \( G \) is the maximum detour distance from \( v \) to a vertex of \( G \). The detour radius \( \text{rad}_D(G) \) of \( G \) is the minimum detour eccentricity among the vertices of \( G \), while the detour diameter \( \text{diam}_D(G) \) of \( G \) is the maximum detour eccentricity among the vertices of \( G \). It is shown that \(\text{rad}_D(G) < \text{diam}_D(G) < 2\text{rad}_D(G)\) for every connected graph \( G \) and that every pair \( a,b \) of positive integers with \( a \leq b \leq 2a \) is realizable as the detour radius and detour diameter of some connected graph. The detour center of \( G \) is the subgraph induced by those vertices of \( G \) having detour eccentricity \( \text{rad}_D(G) \). A connected graph \( G \) is detour self-centered if \( G \) is its own detour center. The detour periphery of \( G \) is the subgraph induced by the vertices of \( G \) having detour eccentricity \( \text{diam}_D(G) \). It is shown that every graph is the detour center of some connected graph. Detour self-centered graphs are investigated. We present sufficient conditions for a graph to be the detour periphery of some connected graph. Several classes of graphs that are not the detour periphery of any connected graph are determined.

In recent work, Corteel and Lovejoy extensively studied overpartitions as a means of better understanding and interpreting various \( q \)-series identities. Our goal in this article is quite different. We wish to prove a number of arithmetic relations satisfied by the overpartition function. Employing elementary generating function dissection techniques, we will prove identities such as

\[
\sum\limits_{n\geq0}\overline{p}\left(8n + 7\right) q^n = 64 \frac{(q^2)_\infty^{22}}{(q)_\infty^{23}}
\]

and congruences such as

\[
\overline{p}(9n+6) \equiv 0 \pmod{8}
\]

where \( \overline{p}(n) \) denotes the number of overpartitions of \( n \).

Let \( G = (V,E) \) be a graph with \( |V| = p \) and \( |E| = q \). The graph \( G \) is total edge-magic if there exists a bijection \( f : V \cup E \to \{1,2,\ldots,p+q\} \) such that for all \( e = (u,v) \in E \), \( f(u) + f(e) + f(v) \) is constant throughout the graph. A total edge-magic graph is called super edge-magic if \( f(V) = \{1,2,\ldots,p\} \). Lee and Kong conjectured that for any odd positive integer \( r \), the union of any \( r \) star graphs is super edge-magic. In this paper, we supply substantial new evidence to support this conjecture for the case \( r = 3 \).

We show that \( \mathbb{Z} \)-cyclic ordered triplewhist and directed triplewhist tournaments on \( p \) elements exist when \( p \equiv 9 \pmod{16} \) is prime.

If \( G = (V,E,F) \) is a finite connected plane graph on \( |V| = p \) vertices, \( |E| = q \) edges and \( |F| = t \) faces, then \( G \) is said to be \( (a, d) \)-face antimagic iff there exists a bijection \( h: E \to \{1,2,\ldots,q\} \) and two positive integers \( a \) and \( d \) such that the induced mapping \( g_h: F \to \mathbb{N} \), defined by \( g_h(f) = \sum\{h(u,v) : \text{edge } (u,v) \text{ surrounds the face } f\} \), is injective and has the image set \( g_h(F) = \{a,a+d,\ldots,a + (t – 1)d\} \). We deal with \( (a,d) \)-face antimagic labelings for a certain class of plane graphs.

We provide tables which summarize various aspects of the finite linear groups \( \text{GL}(n, 2) \), \( n < 7 \), in their action upon the vector space \( V_n = V(n, 2) \) and upon the associated projective space \( \text{PG}(n – 1, 2) \). It is intended that the tabulated results should be immediately accessible to finite geometers, and to all others (design theorists, coding theorists, \ldots) who have occasional need of these groups. In the case \( n = 4 \) attention is also paid to the maximal subgroup \( \Gamma \text{L}(2, 4) \). In the case \( n = 6 \) the maximal subgroups \( \Gamma \text{L}(2, 8) \) and \( \Gamma \text{L}(3, 4) \) are treated, as are class aspects of the tensor product structure \( V_6 = V_2 \otimes V_3 \), and of the exterior product structure \( V_6 = \wedge^2 V_4 \).

It is conjectured that any 2-regular graph \( G \) with \( n \) edges has a \( \rho \)-labeling (and thus divides \( K_{2n+1} \) cyclically). In this note, we show that the conjecture holds when \( G \) has at most two components.