Let \( G = (V, E) \) be a graph with chromatic number \( k \). A dominating set \( D \) of \( G \) is called a chromatic transversal dominating set (ctd-set) if \( D \) intersects every color class of any \( k \)-coloring of \( G \). The minimum cardinality of a ctd-set of \( G \) is called the chromatic transversal domination number of \( G \) and is denoted by \( \gamma_{ct}(G) \). In this paper, we obtain sharp upper and lower bounds for \( \gamma_{ct} \) for the Mycielskian \( \mu(G) \) and the shadow graph \( \text{Sh}(G) \) of any graph \( G \). We also prove that for any \( c \geq 2 \), the decision problem corresponding to \( \gamma_{ct} \) is NP-hard for graphs with \( \chi(G) = c \).

Let \( G(V, E) \) be a simple graph, and let \( f \) be an integer function defined on \( V \) with \( 1 \leq f(v) \leq d(v) \) for each vertex \( v \in V \). An \( f \)-edge covered colouring is an edge colouring \( C \) such that each colour appears at each vertex \( v \) at least \( f(v) \) times. The maximum number of colours needed to \( f \)-edge covered colour \( G \) is called the \( f \)-edge covered chromatic index of \( G \) and denoted by \( \chi_{fc}'(G) \). Any simple graph \( G \) has an \( f \)-edge covered chromatic index equal to \( \delta_f \) or \( \delta_f – 1 \), where \( \delta_f = \min \left\{\left\lfloor\frac{d(v)}{f(v)}\right\rfloor : v \in V(G)\right\} \). Let \( G \) be a connected and not complete graph with \( \chi_{fc}’ = \delta_f – 1 \). If for each \( u, v \in V \) and \( e = uv \notin E \), we have \( \chi_{fc}'(G+e) > \chi_{fc}'(G) \); then \( G \) is called an \( f \)-edge covered critical graph. In this paper, some properties of \( f \)-edge covered critical graphs are discussed. It is proved that if \( G \) is an \( f \)-edge covered critical graph, then for each \( u, v \in V \) and \( e = uv \notin E \) there exists \( w \in \{u, v\} \) with \( d(w) \leq \delta_f(f(w) + 1) – 2 \) such that \( w \) is adjacent to at least \( \max \left\{d(w) – \delta_f f(w) + 1, (f(w) + 2)d(w) – \delta_f(f(w) + 1)^2 + f(w) + 3\right\} \) vertices which are all \( \delta_f \)-vertices in \( G \).

For a connected graph \(G\) of order \(3\) or more and an edge coloring \(c: E(G) \to \mathbb{Z}_k\) (\(k \geq 2\)) where adjacent edges may be colored the same, the color sum \(s(v)\) of a vertex \(v\) of \(G\) is the sum in \(\mathbb{Z}_k\) of the colors of the edges incident with \(v\). The edge coloring \(c\) is a modular \(k\)-edge coloring of \(G\) if \(s(u) \neq s(v)\) in \(\mathbb{Z}_k\) for all pairs \(u, v\) of adjacent vertices in \(G\). The modular chromatic index \(\chi_m'(G)\) of \(G\) is the minimum \(k\) for which \(G\) has a modular \(k\)-edge coloring. It is shown that \(\chi(G) \leq \chi_m'(G) \leq \chi(G)+1\) for every connected graph \(G\) of order at least 3, where \(\chi(G)\) is the chromatic number of \(G\). Furthermore, it is shown that \(\chi_m'(G) = \chi(G) + 1\) if and only if \(\chi(G) \equiv 2 \pmod{4}\) and every proper \(\chi(G)\)-coloring of \(G\) results in color classes of odd size.

It is known that an \(\alpha\)-labeling of a bipartite graph \(G\) with \(n\) edges can be used to obtain a cyclic \(G\)-decomposition of \(K_{2nx+1}\) for every positive integer \(x\). It is also known that if two graphs \(G\) and \(H\) admit a free \(a\)-labeling, then their vertex-disjoint union also admits a free \(\alpha\)-labeling. We show that if \(G\) is a bipartite prism, a bipartite Möbius ladder, or a connected cubic bipartite graph of order at most 14, then \(G\) admits a free \(a\)-labeling. We conjecture that every bipartite cubic graph admits a free \(\alpha\)-labeling.

Here we present a characterization of Sheffer-type polynomial sequences based on the isomorphism between the Riordan group and Sheffer group and the sequence characterization of Riordan arrays. We also give several alternative forms of the characterization of the Riordan group, Sheffer group, and their subgroups. Formulas for the computation of the generating functions of Riordan arrays and Sheffer-type polynomial sequences from the characteristics are shown. Furthermore, the applications of the characteristics to lattice walks and recursive construction of Sheffer-type polynomial sequences are also given.

A set of \( S \) edge-disjoint Hamilton cycles in a graph \( G \) is said to be \({maximal}\) if the Hamilton cycles in \( S \) form a subgraph of \( G \) such that \( G – E(S) \) has no Hamilton cycle. The set of integers \( m \) for which a graph \( G \) contains a maximal set of \( m \) edge-disjoint Hamilton cycles has previously been determined whenever \( G \) is a complete graph, a complete bipartite graph, and in many cases when \( G \) is a complete multipartite graph. In this paper, we solve half of the remaining open cases regarding complete multipartite graphs.

For an outerplanar graph on \( n \) vertices, we determine the maximum number of vertices of degree at least \( k \). For \( k = 4 \) (and \( n \geq 7 \)) the answer is \( n – 4 \). For \( k = 5 \) (and \( n \geq 4 \)), the answer is \( \left\lfloor \frac{2(n-4)}{3} \right\rfloor \) (except one less when \( n \equiv 1 \pmod{6} \)). For \( k \geq 6 \) (and \( n \geq k + 2 \)), the answer is \( \left\lfloor \frac{n-6}{k-4} \right\rfloor \). We also determine the maximum sum of the degrees of \( s \) vertices in an \( n \)-vertex outerplanar graph and the maximum sum of the degrees of the vertices with degree at least \( k \).

For a connected graph \( G \) and a positive integer \( k \), the \( k \)th power \( G^k \) of \( G \) is the graph with \( V(G^k) = V(G) \) where \( uv \in E(G^k) \) if the distance \( d_G(u,v) \) between \( u \) and \( v \) is at most \( k \). The edge coloring of \( G^k \) defined by assigning each edge \( uv \) of \( G^k \) the color \( d_G(u,v) \) produces an edge-colored graph \( G^k \) called a distance-colored graph. A distance-colored graph is properly \( p \)-connected if every two distinct vertices \( u \) and \( v \) in the graph are connected by \( p \) internally disjoint properly colored \( u \)-\( v \) paths. It is shown that \( G^2 \) is properly \( 2 \)-connected for every \( 2 \)-connected graph that is not complete, a double star is the only tree \( T \) for which \( T^2 \) is properly \( 2 \)-connected, and \( G^3 \) is properly \( 2 \)-connected for every connected graph \( G \) of diameter at least \( 3 \). All pairs \( k,n \) of positive integers for which \( P_n^k \) is properly \( k \)-connected are determined. It is shown that every properly colored graph \( H \) with \( \chi'(H) \) colors is a subgraph of some distance-colored graph and the question of determining the smallest order of such a graph is studied.

Let \( G \) be a simple graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( \mathbb{Z}_2 = \{0,1\} \). Any edge labeling \( f \) induces a partial vertex labeling \( f^+ : V(G) \to \mathbb{Z}_2 \) assigning \( 0 \) or \( 1 \) to \( f^+(v) \), \( v \) being an element of \( V(G) \), depending on whether there are more \( 0 \)-edges or \( 1 \)-edges incident with \( v \), and no label is given to \( f^+(v) \) otherwise. For each \( i \in \mathbb{Z}_2 \), let \( v_f(i) = |\{v \in V(G) : f^+(v) = i\}| \) and let \( e_f(i) = |\{e \in E(G) : f(e) = i\}| \). An edge-labeling \( f \) of \( G \) is said to be edge-friendly if \( |e_f(0) – e_f(1)| \leq 1 \). The edge-balance index set of the graph \( G \) is defined as \( EBI(G) = \{|v_f(0) – v_f(1)| : f \text{ is edge-friendly}\} \). In this paper, exact values of the edge-balance index sets of \( L \)-product of cycles with stars, \( C_n \times_L (S_t(m), c) \), where \( m \) is even, and \( c \) is the center of the star graph are presented.

We investigate group divisible designs with two association classes (known as GDDS, GADs or PBIBDs) with block size 3 and unequal size groups. We completely determine the necessary and sufficient conditions for groups with size vector \((n, 1)\) for any \(n \geq 3\), and \((n, 2, 1)\) for \(n \in \{2, 3, \ldots, 6\}\). We also have some general results for \((n_1, n_2, n_3)\).