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The crossing number of a graph is a significant measure that indicates the complexity of the graph and the difficulty of visualizing it. In this paper, we examine the crossing numbers of join products involving the complete graph K 5 with discrete graphs, paths, and cycles. We analyze optimal drawings of K 5 , identify all five non-isomorphic drawings, and address previously hypothesized crossing numbers for K 5 + P n , and K 5 + C n . Through a simplified approach, we first establish cr (K 5 + D n) and then extend our method to prove the crossing numbers cr (K 5 + P n) and cr (K 5 + C n) . These results also lead to new hypotheses for cr (W m + S n) and cr (W m + W n) involving wheels and stars. Our findings correct previous inaccuracies in the literature and provide a foundation for future research. [ABSTRACT FROM AUTHOR]

Circuits in a graph are interesting structures and identifying them is of an important relevance for many applications. However, enumerating circuits is known to be a difficult problem, since their number can grow exponentially. In this paper, we propose fast parallel approaches for enumerating elementary circuits of directed graphs based on graphics processing unit (GPU). Our algorithms are based on a massive exploration of the graph in a breadth-first search strategy. Algorithm V-FEC explores the graph starting from different vertices simultaneously. To further reduce the search space, we present T-FEC, another algorithm that uses triplets as an initial set to start exploring. To the best of our knowledge, those are the first parallel GPU-based algorithms for finding all circuits of a given graph. In addition, they find circuits of a given length and circuits with a specific vertex or edge. The evaluation results show that the proposed approaches achieve up to 190x speed-up over Johnson's algorithm, one of the most efficient sequential algorithms for finding circuits. [ABSTRACT FROM AUTHOR]

Let G be a connected graph and g be a non-negative integer. A vertex set S of graph G is called a g-extra cut if G − S is disconnected and each component of G − S has at least g + 1 vertices. The g-extra connectivity of G is the minimum cardinality of a g-extra cut of G if G has at least one g-extra cut. For two graphs G 1 = (V 1 , E 1) and G 2 = (V 2 , E 2) , the strong product G 1 ⊠ G 2 is defined as follows: its vertex set is V 1 × V 2 and its edge set is { (x 1 , x 2) (y 1 , y 2) | x 1 = x 2 and y 1 y 2 ∈ E 2 ; or y 1 = y 2 and x 1 x 2 ∈ E 1 ; or x 1 x 2 ∈ E 1 and y 1 y 2 ∈ E 2 } , where (x 1 , x 2) , (y 1 , y 2) ∈ V 1 × V 2 . In this paper, we obtain the g-extra connectivity of the strong product of two paths, the strong product of a path and a cycle, and the strong product of two cycles. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate eventual periodicity of the following max-type system of difference equations of higher order with four variables ... [ABSTRACT FROM AUTHOR]

A set S of vertices in a graph G is a dominating set of G if every vertex not in S is adjacent to some vertex in S. The domination number, (G), of G is the minimum cardinality of a dominating set of G. The authors proved in [A new lower bound on the domination number of a graph, J. Comb. Optim. 38 (2019) 721-738] that if G is a connected graph of order n = 2 with k = 0 cycles and l leaves, then (G) = (n - l + 2 - 2k)/3. As a consequence of the above bound, (G) = (n - l + 2(1 - k) + m)/3 for some integer m = 0. In this paper, we characterize the class of cactus graphs achieving equality here, thereby providing a classification of all cactus graphs according to their domination number. [ABSTRACT FROM AUTHOR]

In 2013, Duchene, Kheddouci, Nowakowski and Tahraoui introduced a labeled version of the graph packing problem. It led to the introduction of a new graph parameter, the k-packing label-span k. This parameter corresponds, given a graph H on n vertices, to the maximum number of labels we can assign to the vertices of the graph, such that there exists a packing of k copies of H into the complete graph Kn, coherent with the labels of the vertices. The authors intensively studied the labeled packing of cycles, and, among other results, they conjectured that for every cycle Cn of order n = 2k + x, with k = 2 and 1 = x = 2k 1, the value of k(Cn) was 2 if x is 1 and k is even, and x + 2 otherwise. In this paper, we disprove this conjecture by giving a counterexample. We however prove that it gives a valid lower bound, and we give sufficient conditions for the upper bound to hold. We then give some similar results for the labeled packing of circuits. [ABSTRACT FROM AUTHOR]

Alspach et al. (J. Austral. Math. Soc. 56(3) (1994) 391–402) constructed an infinite family of tetravalent graphs M(a; m, n) and proved that if n ≥ 9 be odd and a 3 ≡ 1 (mod n) , then M(a; 3, n) is half-arc-transitive. In this paper, we show that if a 3 ≡ 1 (mod n) , then M(a; 3, n) is an infinite family of tetravalent half-arc-transitive Cayley graphs for all integers n except 7 and 14. [ABSTRACT FROM AUTHOR]

Recently Alexandersson and Getachew proved some multivariate generalizations of a formula for enumerating signed excedances in derangements. In this paper we first relate their work to a recent continued fraction for permutations and confirm some of their observations. Our second main result is two refinements of their multivariate identities, which clearly explain the meaning of each term in their main formulas. We also explore some similar formulas for permutations of type B. [ABSTRACT FROM AUTHOR]

The paper promotes the notion of computational experiment supported by a multi-tool digital environment as a means of the development of new mathematical knowledge in the context of education. The main study of the paper deals with the issues of teaching this knowledge to secondary teacher candidates within a graduate capstone mathematics education course. The interplay of mathematics and education is considered through the lens of using technology to enhance one's mathematical background by advancing ideas from mostly known to genuinely unknown. In this paper, the knowns consist of Fibonacci numbers, Pascal's triangle, and continued fractions; among the unknowns are Fibonacci-like polynomials and generalized golden ratios in the form of cycles of various lengths. The paper discusses the interplay of pragmatic and epistemic uses of digital tools by the learners of mathematics. The data for the study were collected over the years through solicited comments by teacher candidates enrolled in the capstone course. The main results indicate the candidates' appreciation of the need for deep mathematical knowledge as an instrument of the modern-day pedagogy aimed at making high schoolers interested in the subject matter. [ABSTRACT FROM AUTHOR]