The game of Nim as played on graphs was introduced in \([3]\) and extended in \([4]\) by Masahiko Fukuyama. His papers detail the calculation of Grundy numbers for graphs under specific circumstances. We extend these results and introduce the strategy for even cycles. This paper examines a more general class of graphs by restricting the edge weight to one. We provide structural conditions for which there exist a winning strategy. This yields the solution for the complete graph.

Given positive integers \(j\) and \(k\) with \(j \geq k\), an {L\((j,k)\)-labeling} of a graph \(G\) assigns nonnegative integers to \(V(G)\) such that adjacent vertices’ labels differ by at least \(j\), and vertices distance two apart have labels differing by at least \(k\). The span of an L\((j,k)\)-labeling is the difference between the maximum and minimum assigned integers. The \(\lambda_{j,k}\)-number of \(G\) is the minimum span over all L\((j,k)\)-labelings of \(G\). This paper investigates the \(\lambda_{j,k}\)-numbers of Cartesian products of three complete graphs.

An \(L(2,1)\)-labeling of a graph \(G = (V, E)\) is a function \(f\) from its vertex set \(V\) to the set of nonnegative integers such that \(|f(x) – f(y)| \geq 2\) if \(xy \in E\) and \(|f(x) – f(y)| \geq 1\) if \(x\) and \(y\) are at distance two apart. The span of an \(L(2,1)\)-labeling \(f\) is the maximum value of \(f(x)\) over all \(x \in V\). The \emph{\(L(2,1)\)-labeling number} of \(G\), denoted \(\lambda(G)\), is the least integer \(k\) such that \(G\) has an \(L(2,1)\)-labeling of span \(k\). Chang and Kuo [1996, SIAM J. Discrete
Mathematics, Vol 9, No. 2, pp. \(309 — 316]\) proved that \(\lambda(G) \leq 2\Delta(G)\) and conjectured that \(\lambda(G) \leq \Delta(G) + \omega(G)\) for a strongly chordal graph \(G\), where \(\Delta(G)\) and \(\omega(G)\) are the maximum degree and maximum clique size of \(G\), respectively. In this paper, we propose an algorithm for \(L(2,1)\)-labeling a block graph \(G\) with \(\Delta(G) + \omega(G) + 1\) colors. As block graphs are strongly chordal graphs, our result proves Chang and Kuo’s conjecture for block graphs. We also obtain better bounds of \(\lambda(G)\) for some special subclasses of block graphs. Finally, we investigate finding the exact value of \(\lambda(G)\) for a block graph \(G\).

There are \(267\) nonisomorphic groups of order \(64\). It was known that \(259\) of these groups admit \((64, 28, 12)\) difference sets. In \([4]\), the author found all \((64, 28, 12)\) difference sets in \(111\) groups. In this paper, we find all \((64, 28, 12)\) difference sets in all the remaining groups of order \(64\) that admit \((64, 28, 12)\) difference sets. Also, we find all nonisomorphic symmetric \((64, 28, 12)\) designs that arise from these difference sets. We use these \((64, 28, 12)\) difference sets to construct all \((64, 27, 10, 12)\) and \((64, 28, 12, 12)\) partial difference sets. Finally, we examine the corresponding strongly regular graphs with parameters \((64, 27, 10, 12)\) and \((64, 28, 12, 12)\).

In terms of Sears’ transformation formula for \(_4\phi_3\)-series, we provide standard proofs of a summation formula for \(_4\phi_3\)-series due to Andrews [Andrews, Adv. Appl. Math. \(46 (2011), 15-24]\) and another summation formula for \(_4\phi_3\)-series conjectured in the same paper. Meanwhile, several other related results are also derived.

In the book embedding of an ordered set, the elements of the set are embedded along the spine of a book to form a linear extension. The pagenumber (or stack number) is the minimum number of pages needed to draw the edges as simple curves such that
edges drawn on the same page do not intersect. The pagenumber problem for ordered sets is known to be NP-complete, even if the order of the elements on the spine is-fixed. In this paper, we investigate this problem for some classes of ordered sets. We provide an efficient algorithm for embedding bipartite interval orders in a book with the minimum number of pages. We also give an upper bound for the pagenumber of general bipartite ordered sets and the pagenumber of complete multipartite ordered sets. At the end of this paper we discuss the effect of a number of diagram operations on the pagenumber of ordered sets. We give an answer to an open question by Nowakowski and Parker \([7]\) and we provide several known and new open questions we consider worth investigating.

Let \(\Gamma\) be a \(d\)-bounded distance-regular graph with diameter \(d \geq 2\).In this paper, we give some counting formulas of subspaces in \(\Gamma\) and construct an authentication code with perfect. secrecy.

We determine the full friendly index sets of spiders and disprove a conjecture by Lee and Salehi \([4]\) that the friendly index set of a tree forms an arithmetic progression.

Let \(k\) be a positive integer and \(G = (V(G), E(G))\) a graph. A subset \(S \subseteq V(G)\) is a \(k\)-dominating set if every vertex of \(V(G)- S\) is adjacent to at least \(k\) vertices of \(S\). The \(k\)-domination number \(\gamma_k(G)\) is the minimum cardinality of a \(k\)-dominating set of \(G\). A graph \(G\) is called \(\gamma_k\)-stable if \(\gamma_{\bar{k}}(G – e) = \gamma_{{k}}(G)\) for every edge \(e\) of \(E(G)\). We first provide a necessary and sufficient condition for \(\gamma_{\bar{k}}\)-stable graphs. Then, for \(k \geq 2\), we offer a constructive characterization of \(\gamma_{\bar{k}}\)-stable trees.

The zero-divisor graph of a commutative semigroup with zero is a graph whose vertices are the nonzero zero-divisors of the semigroup, with two distinct vertices joined by an edge if their product in the semigroup is zero. In this paper, we provide formulas to calculate the numbers of non-isomorphic zero-divisor semigroups corresponding to star graphs \(K_{1,m}\), two-star graphs \(T_{m,n}\), and windmill graphs, respectively.