A union closed (UC) family \( \mathcal{A} \) is a finite family of sets such that the union of any two sets in \( \mathcal{A} \) is also in \( \mathcal{A} \). Peter Frankl conjectured in 1979 that for every union closed family \( \mathcal{A} \), there exists some \( x \) contained in at least half the members of \( \mathcal{A} \). In this paper, we show that if a UC family \( \mathcal{A} \) fails the conjecture, then no element can appear in more than two of its \( 3 \)-sets, and so the number of \( 3 \)-sets in \( \mathcal{A} \) can be no more than \( \frac{2n}{3} \).

A \( (k,d) \)-total coloring (\( k,d \in \mathbb{N}, k \geq 2d \)) of a graph \( G \) is an assignment \( c \) of colors \( \{0,1,\ldots,k-1\} \) to the vertices and edges of \( G \) such that \( d \leq |c(x_i) – c(x_j)| \leq k – d \) whenever \( x_i \) and \( x_j \) are two adjacent edges, two adjacent vertices, or an edge incident to a vertex. The circular total chromatic number \( \chi_c”(G) \) is defined by

\[\chi_c”(G) = \inf\{k/d : G \text{ has a } (k, d)\text{-total coloring}\}.\]

It was proved that \( \chi”(G) – 1 < \chi_c''(G) \leq \chi''(G) \) — where \( \chi''(G) \) is the total chromatic number of \( G \) — with equality for all type-1 graphs and most of the so far considered type-2 graphs. We determine an infinite class of graphs \( G \) such that \( \chi_c''(G) < \chi''(G) \) and we list all graphs of order \( <7 \) with this property.

We first prove that if \( G \) is a connected graph with \( n \) vertices and chromatic number \( \chi(G) = k \geq 2 \), then its independent domination number

\[i(G) \leq \left\lceil \frac{(k-1)}{k}n \right\rceil – (k-2).\]

This bound is tight and remains so for planar graphs. We then prove that the independent domination number of a diameter two planar graph on \( n \) vertices is at most \( \left\lceil \frac{n}{3} \right\rceil \).

We first prove that if \( G \) is a connected graph with \( n \) vertices and chromatic number \( \chi(G) = k \geq 2 \), then its independent domination number

\[i(G) \leq \left\lceil \frac{(k-1)}{k}n \right\rceil – (k-2).\]

This bound is tight and remains so for planar graphs. We then prove that the independent domination number of a diameter two planar graph on \( n \) vertices is at most \( \left\lceil \frac{n}{3} \right\rceil \).

Hill, Landjev, Jones, Storme, and Barat proved in a previous article on caps in PG(5, 3) and PG(6,3) that every 53-cap in PG(5, 3) is contained in the 56-cap of Hill and that there exist complete 48-caps in PG(5,3). The first result was used to lower the upper bound on \( m_2(6,3) \) on the size of caps in PG(6, 3) from 164 to 154. Presently, the known upper bound on \( m_2(6, 3) \) is 148. In this article, using computer searches, we prove that every 49-cap in PG(5, 3) is contained in a 56-cap, and that every 48-cap, having a 20-hyperplane with at most 8-solids, is also contained in a 56-cap. Computer searches for caps in PG(6,3) which use the computer results of PG(5,3) then lower the upper bound on \( m_2(6,3) \) to \( m_2(6,3) \leq 136 \). So now we know that \( 112 \leq m_2(6,3) \leq 136 \).

Let \( \delta(G) \) and \( \lambda(G) \) be the minimum degree and edge-connectivity of a graph \( G \), respectively. A graph \( G \) is maximally edge-connected if \( \lambda(G) = \delta(G) \) and super-edge-connected if every minimum edge cut consists of edges adjacent to a vertex of minimum degree.

In this paper, sufficient conditions for super-edge-connected graphs depending on the clique number and the minimum degree are presented. These results show that some known sufficient conditions for maximally edge-connected graphs even lead to super-edge-connected graphs.