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.