A \( d \)-angulation of a surface is an embedding of a 3-connected graph on that surface that divides it into \( d \)-gonal faces. A \( d \)-angulation is said to be Grünbaum colorable if its edges can be \( d \)-colored so that every face uses all \( d \) colors. Up to now, the concept of Grünbaum coloring has been related only to triangulations (\( d = 3 \)), but in this note, this concept is generalized for an arbitrary face size \( d \geq 3 \). It is shown that the face 2-colorability of a \( d \)-angulation \( P \) implies the Grünbaum colorability of \( P \). Some wide classes of triangulations have turned out to be face 2-colorable.

Let \( G \) be a claw-free graph of order \( 4k \), where \( k \) is a positive integer. In this paper, it is proved that if the degree sum \( d(u) + d(v) \) is at least \( 4k – 2 \) for every pair of nonadjacent vertices \( u, v \in V(G) \), then \( G \) has a spanning subgraph consisting of \( k – 1 \) quadrilaterals and a 4-path such that all of them are vertex-disjoint, unless \( G \) is isomorphic to \( K_{4k_1 + 2} \cup K_{4k_2 + 2} \), or \( K_{4k_1 + 1} \cup K_{4k_2 + 3} \), where \( k_1 \geq 0, k_2 \geq 0, k_1 + k_2 = k – 1 \). We further showed that the requirement about claw-free is indispensable and the degree condition is sharp.

For \( n \geq 1 \) we call a sequence \( s_1, s_2, \ldots, s_n \) an up-down sequence of length \( n \) when (i) \( s_1 = 1 \); (ii) \( s_i \in \{1, 2, 3, 4\} \), for \( 2 \leq i \leq n \); and, (iii) \( |s_i – s_{i-1}| = 1 \), for \( 2 \leq i \leq n \). We count the number of inversions and coinversions for all such up-down sequences of length \( n \), as well as the sum of the major indices for all these sequences of length \( n \).

Das \([4]\), Feng et al. \([5]\), and Li et al. \([13]\) obtained upper bounds for the number of spanning trees of a connected graph. Using some ideas in \([4]\), \([5]\), and \([13]\) and other established results, we obtain new upper bounds for the number of spanning trees of a connected graph.

Given two non-isomorphic bipartite 2-factors \( F_1 \) and \( F_2 \) of order \( 4n \), the Bipartite Hamilton-Waterloo Problem (BHWP) asks for a 2-factorization of \( K_{2n,2n} \) into \( \alpha \) copies of \( F_1 \) and \( \beta \) copies of \( F_2 \), where \( \alpha + \beta = n \) and \( \alpha, \beta \geq 1 \). We show that the BHWP has a solution when \( F_1 \) is a refinement of \( F_2 \), where no component of \( F_1 \) is a \( C_4 \) or \( C_6 \), except possibly when \( \alpha = 1 \) and either (i) \( F_2 \) is a \( C_4 \)-factor or (ii) \( F_2 \) has more than one \( C_4 \) with all other components of an order \( r \equiv 0 \pmod{4} > 4 \) or (iii) \( F_2 \) has components with an order \( r \equiv 2 \pmod{4} \), when \( n \) is even. We also show that there does not exist a factorization of \( K_{6,6} \) into a single 12-cycle and two \( C_4 \)-factors.

For every integer \( c \), let \( r(2, 2, c) \) be the least integer \( n \) such that for every 2-coloring of the set \( \{1, 2, \ldots, n\} \) there exists a monochromatic solution to the equation \[ 2x_1 + 2x_2 + c = x_3. \]
Secondly, for every integer \( c \), let \( r(2, 2, 2, c) \) be the least integer \( n \) such that for every 2-coloring of the set \( \{1, 2, \ldots, n\} \) there exists a monochromatic solution to the equation \[ 2x_1 + 2x_2 + 2x_3 + c = x_4. \]
In this paper, exact values are found for \( r(2, 2, c) \) and \( r(2, 2, 2, c) \).

We construct a class of maximal partial line spreads in \( \mathrm{PG}(4, q) \), that we call \( q \)-added maximal partial line spreads. We obtain them by depriving a line spread of a hyperplane of some lines and adding \( q+1 \) pairwise skew lines not in the hyperplane for each removed line. We do it in a theoretical way for every value of \( q \), and by a computer search for \( q \leq 16 \). More precisely, we prove that for every \( q \) there are \( q \)-added MPS of size \( q^2 + kq + 1 \), for every integer \( k = 1, \ldots, q-1 \), while by a computer search we get larger cardinalities.

Let \( i(G) \) denote the minimum cardinality of an independent dominating set for \( G \). A graph \( G \) is \( k \)-\( i \)-critical if \( i(G) = k \), but \( i(G + uv) < k \) for any pair of non-adjacent vertices \( u \) and \( v \) of \( G \). In this paper, we show that if \( G \) is a connected \( k \)-\( i \)-critical graph, for \( k \geq 3 \), with a cutvertex \( u \), then the number of components of \( G – u \), \( \omega(G – u) \), is at most \( k – 1 \) and there are at most two non-singleton components. Further, if \( \omega(G – u) = k – 1 \), then a characterization of such graphs is given.

For a nontrivial connected graph \( G \), let \( c: V(G) \to \mathbb{Z}_2 \) be a vertex coloring of \( G \) where \( c(v) \neq 0 \) for at least one vertex \( v \) of \( G \). Then the coloring \( c \) induces a new coloring \( \sigma: V(G) \to \mathbb{Z}_2 \) defined by \( \sigma(v) = \sum_{u \in N[v]} c(u) \), where \( N[v] \) is the closed neighborhood of \( v \) and addition is performed in \( \mathbb{Z}_2 \). If \( \sigma(v) = 0 \in \mathbb{Z}_2 \) for every vertex \( v \) in \( G \), then the coloring \( c \) is called a modular monochromatic \( (2, 0) \)-coloring of \( G \). A graph \( G \) having a modular monochromatic \( (2, 0) \)-coloring is a monochromatic \( (2, 0) \)-colorable graph. The minimum number of vertices colored 1 in a modular monochromatic \( (2, 0) \)-coloring of \( G \) is the \( (2, 0) \)-chromatic number \( \chi_{(2,0)}(G) \) of \( G \). A monochromatic \( (2, 0) \)-colorable graph \( G \) of order \( n \) is \( (2, 0) \)-extremal if \( \chi_{(2,0)}(G) = n \). It is known that a tree \( T \) is \( (2,0) \)-extremal if and only if every vertex of \( T \) has odd degree. In this work, we characterize all trees of order \( n \) having \( (2,0) \)-chromatic number \( n-1 \), \( n-2 \), or \( n-3 \), and investigate the structures of connected graphs having large \( (2, 0) \)-chromatic numbers.

A seed of a word \( x \) is a cover of a superword of \( x \). In this paper, we study the frequency of appearance of seeds in words. We give bounds for the average number of seeds in a word and we investigate the maximum number of distinct seeds that can appear in a word. More precisely, we prove that a word has \( O(n) \) seeds on average and that the maximum number of distinct seeds in a word is between \( \frac{1}{6}(n^2) + o(n^2) \) and \( \frac{1}{4}(n^2) + o(n^2) \), and we reveal some properties of an extremal word for the last case.