By eliminating the win condition in the game of Connect Four and extending the board to infinite height, a rich state space of positions is obtained. We investigate the number of positions reachable on an \(n\)-column board after \(k\) color-alternating moves. For fixed \(k\) we demonstrate polynomiality, derive a partial formula for the polynomial coefficients, and precisely characterize the asymptotic behavior as \(n \to \infty\). We then turn our attention to the fixed-\(n\) case and show that, under a natural addition operation, positions reachable in an even number of moves form a monoid with a highly symmetric finite generating set; by examining certain free submonoids, we bound the exponential growth rate as \(k \to \infty\).

Sterboul’s theorem characterizes non-Kőnig–Egerváry graphs by the presence, relative to a maximum matching, of a flower or a posy. In this paper we translate that obstruction into the language of perfect flowers and the core of the graph. We introduce core-defective perfect flowers: perfect flowers whose alternating path contains a vertex at odd distance from the blossom base that does not belong to the core. We prove first that every Kőnig–Egerváry graph is core-rigid: in every perfect flower, all odd-distance vertices of the attaching path lie in the core. Conversely, if \(G\) is connected and is not an odd cycle, then \(G\) is non-Kőnig–Egerváry if and only if \(G\) contains a core-defective perfect flower. Thus, among connected graphs different from an odd cycle, the Kőnig–Egerváry graphs are exactly the graphs with no core-defective perfect flower. In the matchable case the statement strengthens: if \(G\) has a perfect matching, then being non-Kőnig–Egerváry is equivalent to the existence of a core-defective perfect flower for some maximum matching, and also equivalent to the existence of one for every maximum matching. We include examples and counterexamples showing why odd cycles, disconnected graphs, and the universal quantifier over maximum matchings require separate treatment.

A hypergraph \(H\) is said to be \(r\)-partite \(r\)-uniform if its vertex set \(V\) can be partitioned into non-empty sets \(V_1, V_2, \cdots, V_r\) so that every edge in the edge set \(E(H)\), consists of precisely one vertex from each set \(V_i\), \(i=1,2,\cdots,r\). It is denoted as \(H^r(V_1,V_2,\cdots,V_r)\) or \(H^r_{(n_1,n_2,\cdots,n_r)}\) if \(|V_i|=n_i\) for \(i=1,2,\cdots,r\). There exists an \(r\)-partite self-complementary \(r\)-uniform hypergraph \(H^r(V_1,V_2,\cdots,V_r)\) where \(|V_i|=n_i\) for \(i=1,2,\cdots,r\) if and only if at least one of \(n_1,n_2,\cdots,n_r\) is even. And there exists an \(r\)-partite almost self-complementary \(r\)-uniform hypergraph \(H^r(V_1, V_2,\cdots,V_r)\) where \(|V_i|=n_i\) for \(i=1,2,\cdots,r\) if and only if \(n_1,n_2,\cdots,n_r\) are odd. In this paper, we prove the existence of regular \(3\)-partite self-complementary \(3\)-uniform hypergraphs. Further we prove there does not exist a regular \(3\)-partite almost self-complementary \(3\)-uniform hypergraph.

We study the difference between the numbers of even and odd permutations in \(\mathfrak{S}_n\) having exactly \(k\) fixed points. We derive a closed formula for this quantity using four complementary approaches: exponential generating functions, a determinant representation, a combinatorial derivation based on inclusion–exclusion on cycle structures, and a factorization via the stabilizer subgroup, through restriction to the complement of the fixed-point set. The resulting expression provides a signed refinement of the classical rencontres numbers and yields a simple polynomial form for the associated signed fixed-point distribution.

The present paper gives a detailed study of the structural theory of triple \(\theta\)-skew cyclic codes where the codes are over \(\mathbb{F}_q\). We give a complete characterization of these codes, focusing on their representation as modules. We identify the generator polynomials for the triple skew-cyclic codes as well as those of their duals. We explore the properties of these generator polynomials and their relationship to the code’s structure. Additionally,To illustrate our approach, we give concrete instances of triple \(\theta\)-skew cyclic codes to demonstrate how these structures can behave in practice. The special instances we present reveal that such codes are capable of achieving strong parameters under certain conditions.

Let \(P_{k+1}\) denote a path of length \(k\), let \(S_{m}\) denote a star with \(m\) edges, and let \(K_{n}(\lambda)\) denote the complete multigraph on \(n\) vertices in which every edge is taken \(\lambda\) times. In this paper, we prove that the necessary conditions are also sufficient for a \(\{P_{4}, S_{4}\}\)-decomposition of \(K_{n}(\lambda)\).

A \(\{P_5^{\alpha}, S_5^{\beta}, Y_5^{\gamma}\}\)-decomposition of a graph is a partition of its edge set into \(\alpha\) copies of \(P_5\), \(\beta\) copies of \(S_5\), and \(\gamma\) copies of \(Y_5\), where \(P_5\), \(S_5\), and \(Y_5\) denote the three non-isomorphic trees of order five. In this paper, we study the existence of a \(\{P_5^{\alpha}, S_5^{\beta}, Y_5^{\gamma}\}\)-decomposition of the complete bipartite graph \(K_{m,n}\) for \(m\geq 4\) and \(n\geq 2\), and of the complete graph \(K_n\) for \(n\geq 8\). In fact, we establish necessary and sufficient conditions for the existence of such decompositions in \(K_{m,n}\) and \(K_n\).

Given a connected graph \(G\) and a configuration \(D\) of pebbles on \(V(G)\), a pebble move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. A monophonic path is a longest chordless path between two non-adjacent vertices \(u\) and \(v\). The line segment that connects two vertices on a curve is known as a chord. The monophonic distance between \(u\) and \(v\) is the number of vertices in the longest \(u\)–\(v\) monophonic path, denoted by \(d_{\mu}(u,v)\) in \(G\). The monophonic pebbling number (MPN) of \(G\) is the least number of pebbles needed to guarantee that, from any distribution of pebbles on a graph \(G\), one pebble can be moved to any specified vertex using monophonic paths through pebbling moves. The monophonic \(t\)-pebbling number (MtPN) of \(G\) is the least number of pebbles needed to guarantee that, from any distribution of pebbles, \(t\) pebbles can be moved to any specified vertex using monophonic paths. In this article, we determine the \(MPN\) and \(MtPN\) of Dutch windmill graphs, square of cycles, tadpole graphs, lollipop graphs, double star path graphs, and fuse graphs, and we also discuss their \(t\)-pebbling versions.

In this paper, we consider circulant graphs obtained from the complete graph \(K_N\) by deleting all edges belonging to a prescribed distance class. We study, in a unified manner, the effective resistance, the expected hitting time, the number of spanning trees, and the number of two-component spanning forests of these graphs. For general distance-class deletions, these quantities admit natural spectral representations in terms of the Laplacian eigenvalues. However, such representations typically remain at the level of finite Fourier sums, and concise closed forms are not expected in general. We focus on the case of a single deleted distance class. When the number of vertices \(N\) is odd and \(\gcd(r,N)=1\), the graph \(G_{N,r}\) is isomorphic to \(G_{N,1}\). In this setting, we derive explicit exponential-type formulas for the effective resistance and the number of spanning trees, and obtain corresponding closed expressions for two-component spanning forests and expected hitting times. Our results show that the case \(r=2\) is not essentially new, but follows from a general isomorphism structure underlying distance-class deletions. We also clarify the relation of our formulas to earlier results on the complete graph with a Hamiltonian cycle removed, and provide a unified derivation within a spectral framework. Moreover, by asymptotic analysis, we show that the ratio \(\tau(G_{N,1})/\tau(K_N)\) converges to \(e^{-2}\) as \(N \to \infty\).

A graph \(G\) with vertex set \(V(G)\) and edge set \(E(G)\) is said to have an odd prime labeling if there exists a bijection \(f:V(G)\to \{1,3,5,\dots,2n-1\}\), where \(n=|V(G)|\), such that \(\gcd(f(x),f(y))=1\) for every edge \(xy\in E(G)\). In this paper, we study odd prime labelings of graphs arising from duplication operations on graph elements. We obtain several results for graphs derived from the path graph \(P_n\), the cycle graph \(C_n\), and the star graph \(K_{1,n}\) under various vertex- and edge-duplication constructions.