For a graph \(G\), a \({trail}\) is a vertex-edge alternating sequence \(v_0, e_1, v_1, e_2, \ldots, e_{k-1},v_{k-1}, e_k, v_k\) such that all \(e_i\)’s are distinct and \(e_i = v_{i-1}v_i\) for all \(i\). For \(u, v \in V(G)\), a \((u,v)\)-trail of \(G\) is a trail in \(G\) originating at \(u\) and terminating at \(v\). A closed trail is a \((u,v)\)-trail with \(u = v\). A trail \(H\) is a spanning trail of \(G\) if \(V(H) = V(G)\). Let \(X \subseteq E(G)\) and \(Y \subseteq E(G)\) with \(X \cap Y = \emptyset\). This paper studies the minimum edge-connectivity of \(G\) such that for any \(u, v \in V(G)\) (including \(u = v\)), \(G\) has a spanning \((u, v)\)-trail \(H\) with \(X \subseteq E(H)\) and \(Y \cap E(H) = \emptyset\).

In this paper we settle a long-standing open problem. We prove that it
is \(NP\)-hard to recognize \(T\)-tenacious graphs for any fixed positive rational
number \(T\)

In this paper, we deal with the transitive relations on a finite $n$-element set. The transitive relations are interpreted as Boolean matrices. A special class of transitive relations are constructed and enumerated, which can generate all transitive
relations on a finite n-element set by intersection operation. Besides, several necessary and sufficient conditions that a relation
\(R\) is transitive are given.

In this paper, we obtain an upper bound on the order of a blockwise-burst \([11]\) that can be detected by a row-cyclic array code \([10]\) and obtain the fraction of blockwise-bursts of order exceeding the upper bound that go undetected. We also give a decoding algorithm for the correction of blockwise-bursts in row-cyclic array codes.

In this paper we study defensive alliances in some specific regular graphs, the circulant graphs, i.e. Cayley graphs on a cyclic group.The critical defensive alliances of a circulant graph of degree at most \(6\) are completely determined. For the general case, we give tight lower and upper bounds on the alliance number of a circulant graph with \(d\) generators.

The maximum number of non-isomorphic one-edge extensions \(M(t, n, f)\) of a graph of size \(t\), order \(n\), and vertex degree bounded by \(f\) for \(3 \leq f \leq n-2\) is considered. An upper bound for \(M(t, n, f)\) is obtained, and for the case \(f = n-2\), the exact value is given. Tables are provided for all values of \(M(t, n, f)\) for up to \(n = 12\), \(\left\lfloor \frac{f-1}{2} \right\rfloor < t \leq \left\lfloor \frac{nf}{2} \right\rfloor\), and \(3 \leq f \leq n-2\). Additionally, the relation of these results to the transition digraph for the Random \(f\)-Graph Process, a Markov process concerning graphs with vertex degree bounded by \(f\), is noted.

In this paper, we characterize all spanning trees of the \(r\)-cyclic graph \(G_{n,r}\). We provide the formulation of \(f\)-vectors associated with spanning simplicial complexes \(\Delta_s(G_{n,r})\) and, consequently, deduce a formula for computing the Hilbert series of the Stanley-Reisner ring \(k[\Delta_s(G_{n,r})]\). For the facet ideal \(I(\Delta(G_{n,r}))\), we characterize all associated primes. Specifically, for uni-cyclic graphs with cycle length \(m_i\), we prove that the facet ideal \(I(\Delta(G_{n,1}))\) has linear quotients with respect to its generating set. Furthermore, we establish that projdim \((I_{\mathcal{F}}(\Delta_s(G_{n,1}))) = 1\) and \(\beta_i(I_{\mathcal{F}}(\Delta_(G_s{n,1}))) = m_i\) for \(i \leq 1\).

We consider the one-player game called Dundee, where a deck consists of \(s_i\) cards of value \(i\), for \(i = 1, \ldots, v\), and an integer \(m \leq s_1 + \cdots + s_v\). Over \(m\) rounds, the player names a number between \(1\) and \(v\) and draws a random card from the deck, losing if the named number matches the drawn value in at least one round. The famous Problem of Thirteen, proposed by Montmort in 1708, asks for the winning probability when \(v = 13\), \(s_1 = \cdots = s_{13} = 4\), \(m = 13\), and the player names the sequence \(1, \ldots, 13\). Studied by mathematicians including J. and N. Bernoulli, De Moivre, Euler, and Catalan, this problem’s strategic aspects remain unexplored. We investigate two variants: one where the player’s Round \(i\) bid depends on previous rounds’ drawn values, which we completely solve, and another where the player must specify all \(m\) bids in advance, solving this for \(s_1 = \cdots = s_v\) and arbitrary \(m\).

Let \(n\) be a positive integer with \(n\geq 2\) and \([n] := \{1, 2, \ldots, n\}\). An \(m\)-partial injective map of \([n]\) is a pair \((A, f)\), where \(A\) is an \(m\)-subset of \([n]\) and \(f: A \rightarrow [n]\) is an injective map. Let \(P =L \cup \{I\}\), where \(L\) is the set of all partial injective maps of \([n]\). Partially ordering \(P\) by ordinary or reverse inclusion yields two families of finite posets. This article proves that these posets are atomic lattices, discusses their geometricity, and computes their characteristic polynomials.

In this paper we study defensive alliances in some regular graphs. We determine which subgraphs could a critical defensive alliance of a graph \(G\) induce, if \(G\) is \(6\)-regular and the cardinality of the alliance is at most \(8\).