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.

In this paper we consider a variation of the classical Turán-type extremal problems. Let \( S \) be an \( n \)-term graphical sequence, and \( \sigma(S) \) be the sum of the terms in \( S \). Let \( H \) be a graph. The problem is to determine the smallest even \( l \) such that any \( n \)-term graphical sequence \( S \) having \( \sigma(S) \geq l \) has a realization containing \( H \) as a subgraph. Denote this value \( l \) by \( \sigma(H, n) \). We show \(\sigma(C_{2m+1}, n) = m(2n – m – 1) + 2, \quad \text{for } m \geq 3, n \geq 3m;\) \(\sigma(C_{2m+2}, n) = m(2n – m – 1) + 4, \quad \text{for } m \geq 3, n \geq 5m – 2. \)

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.

For some fixed \( n_0 \geq 0 \), we study the minimum number of vertices or edges that have to be removed from a graph such that no component of the rest has more than \( n_0 \) vertices.

An \( [r, s, n, t] \)-configuration is a collection \(C\) of \(r\)-sets in \( \{1, \ldots, n\} \) such that every \( s \)-set in \( \{1, \ldots, n\} \) contains at most \( t \) of the \( r \)-sets in \( C \). Studying this generalization of the Steiner system was suggested by a theorem of Poonen on union-closed families of sets. In this paper, we consider only \( [3, 4, n, 2] \)-configurations, and refer to them as \(n\)-configurations; by an \( (n, k) \)-configuration we mean an \(n\)-configuration containing exactly \(k\) \(3\)-sets. An \((n,k)\)-configuration is maximal if it is not contained in any \( (n, k + 1) \)-configuration; finally, \( L(n) \) is the largest integer \(k\) for which an \((n, k)\)-configuration exists. In this paper, we determine \(L(n)\) for \( 4 \leq n \leq 9 \), and characterize all the maximal \( n \)-configurations for \(n = 4, 5,\) and \(6\), as well as the \((n, L(n))\)-configurations for \( n = 7, 8, \) and \( 9 \).

Let \( G \) be a simple graph with vertex set \( V \) and edge set \( E \). A vertex labeling \( f: V \to \{0,1,2\} \) induces an edge labeling \( \bar{f}: E \to \{0,1,2\} \) defined by \( \bar{f}(uv) = |f(u) – f(v)| \). Let \( u_f(i) \) denote the number of vertices \( v \) with \( f(v) = i \), \( i = 0,1,2 \). Similarly, \( e_f(i) \) denotes the number of edges \( uv \) with \( \bar{f}(uv) = i \), \( i = 0,1,2 \). A graph is said to be \( 3 \)-equitable if there exists a vertex labeling \( f \) such that \( |v_f(i) – v_f(j)| \leq 1 \) and \( |e_f(i) – e_f(j)| \leq 1 \) for all \( i \neq j \), \( i, j = 0,1,2 \). In which case, \( f \) is called a \( 3 \)-equitable labeling.

In this paper, we prove that the following graphs are three equitable: (1) Helm graph \( H_n \) (\( n \geq 4 \)), (2) A Flower graph \( FL_n \), (3) One point union \( H_n^{(k)} \) of \( k \)-copies of \( H_n \), \( k \geq 1 \), (4) One point union \( K_4^{(k)} \) of \( k \) copies of \( K_4 \), (5) A \( K_4 \)-snake of \( n \) blocks, each equal to \( K_4 \), (6) A \( C_t \)-snake of \( n \) blocks, \( t = 4,6 \) and \( t = 5 \) with \( n \) not congruent to \( 3 \) modulo \( 6 \).

A defensive alliance in a graph \( G = (V,E) \) is a set of vertices \( S \subseteq V \) satisfying the condition that every vertex \( v \in S \) has at most one more neighbor in \( V – S \) than it has in \( S \). Because of such an alliance, the vertices in \( S \), agreeing to mutually support each other, have the strength of numbers to be able to defend themselves from the vertices in \( V – S \). In this paper, we introduce this new concept, together with a variety of other kinds of alliances, and initiate the study of their mathematical properties.

The distance-\( k \) domination number of graph \( G \), \( \gamma_{\leq k}(G) \), is the cardinality of a smallest set of vertices, \( S \), such that every vertex not in \( S \) is no more than distance \( k \) from at least one vertex of \( S \). Carrington, Harary, and Haynes showed \( |V^0| \geq 2|V^+| \) where \( V^0 = \{u \in V: \gamma_{\leq 1}(G-v) = \gamma_{\leq 1}(G)\} \) and \( V^+ = \{v \in V: \gamma_{\leq 1}(G-v) > \gamma_{\leq 1}(G)\} \). This paper extends the result to distance-\( k \) domination, with the obvious change in definition of \( V^0 \) and \( V^+ \), to show \( |V^0| \geq \frac{2}{2k-1}|V^+| \). Extremal graphs are characterized when \( k = 1 \) and some progress is mentioned on the characterization problem when \( k > 1 \).