We introduce the notion of a move graph, that is, a directed graph whose vertex set is a \(\mathbb Z\)-module \(\mathbb Z_n^m\), and whose arc set is uniquely determined by the action \(M\!:\!\mathbb Z_n^m\to \mathbb Z_n^m\) where \(M\) is an \(m\times m\) matrix with integer entries. We study the manner in which properties of move graphs differ when one varies the choice of cyclic group \(\mathbb Z_n\). Our principal focus is on a special family of such graphs, which we refer to as “sub-add move graphs.”

Given functions \(f,g: [n] \rightarrow [n]\), do there exist \(n\) points \(A_1,A_2,\ldots,A_n\) in some metric space such that \(A_{f(i)},A_{g(i)}\) are the points closest and farthest from point \(A_i\)? In this paper we characterize precisely which pairs of functions have this property. Define \(m(k)\) to be the maximum integer such that any pair of functions \(f,g:[m(k)]\rightarrow [m(k)]\) realizable in some metric space is also realizable in \(\mathbb{R}^k\). We show that \(m(k)\) grows exponentially in \(k\). This answers a question of Croft. We also discuss what happens when looking at minimum and maximum distances separately.

Suppose \(G\) is a graph and \(L\) is a list assignment for \(G\). A request of \(L\) is a function \(r\) with nonempty domain \(D\subseteq V(G)\) such that \(r(v) \in L(v)\) for each \(v \in D\). The triple \((G,L,r)\) is \(\epsilon\)-satisfiable if there exists a proper \(L\)-coloring \(f\) of \(G\) such that \(f(v) = r(v)\) for at least \(\epsilon|D|\) vertices in \(D\). We say \(G\) is \((k, \epsilon)\)-flexible if \((G,L’,r’)\) is \(\epsilon\)-satisfiable whenever \(L’\) is a \(k\)-assignment for \(G\) and \(r’\) is a request of \(L’\). It is known that a graph \(G\) is not \((k, \epsilon)\)-flexible for any \(k\) if and only if \(\epsilon > 1/ \rho(G)\) where \(\rho(G)\) is the Hall ratio of \(G\). The list flexibility number of a graph \(G\), denoted \(\chi_{\ell flex}(G)\), is the smallest \(k\) such that \(G\) is \((k,1/ \rho(G))\)-flexible. A fundamental open question on list flexibility numbers asks: Is there a graph with list flexibility number greater than its coloring number? In this paper, we show that the list flexibility number of any complete multipartite graph \(G\) is at most the coloring number of \(G\). We also initiate the study of list epsilon flexibility functions of complete bipartite graphs which was first suggested by Kaul, Mathew, Mudrock, and Pelsmajer in 2024. Specifically, we completely determine the list epsilon flexibility function of \(K_{m,n}\) when \(m \in \{1,2\}\) and establish some additional bounds for small \(m\). Our proofs reveal a connection to list coloring complete bipartite graphs with asymmetric list sizes which is a topic that was explored by Alon, Cambie, and Kang in 2021.

A \(k\)-edge coloring \(c\) of the edge set \(E (G)\) of a graph \(G\) is a surjective mapping \(c : E (G) \to [k] = \{1, 2, \ldots, k\}\). If \(\mathcal{F}\) and \(\mathcal{H}\) are families of graphs, \(MRS(K_n; \mathcal{F}, \mathcal{H})\) is the set of numbers \(k\) such that there is a \(k\)-edge coloring of \(K_n\) with respect to which there is neither a monochromatic copy of any \(F \in \mathcal{F}\) nor a rainbow copy of any \(H \in \mathcal{H}\) in \(K_n\). Our main result is that for all \(n \geq 2\), \(MRS(K_n;\{\text{odd cycles}\},\{\text{cycles}\}) = \{\lceil \log_2 n \rceil, \ldots, n – 1\}\). The proof will exploit an idea for edge-coloring connected graphs so as to forbid rainbow cycles to be found in [4].

If \(S\) is a numerical semigroup, we will denote by \({\mathrm F}(S),\) \({\mathrm g}(S)\) and \({\mathrm t}(S),\) the Frobenius number, the genus and the type of \(S,\) respectively. We will also denote by \({\mathrm n}(S)\) and \({\mathrm i}(S)\) the cardinality of the sets \(\{s\in S\mid s<{\mathrm F}(S)\}\) and \(\{x\in \mathbb{N}\backslash S\mid x-1\in S\},\) respectively. In this paper we will study the \(\mathrm{PTT}\)-semigroups. That is, perfect numerical semigroups with type two. In particular, we will see that if \(S\) is a numerical semigroup, then the following conditions are equivalent: 1) \(S\) is a \(\mathrm{PTT}\)-semigroup; 2) The set of pseudo-Frobenius numbers of \(S\) is \(\{{\mathrm F}(S),{\mathrm F}(S)-1\}\); 3) \(S\) is maximal in the set \(\{T\mid T \mbox{ is a numerical semigroup } T\cap \{{\mathrm F}(S),{\mathrm F}(S)-1\}=\emptyset \mbox{ and } {\mathrm t}(T)=2\}\); and 4) \({\mathrm F}(S)-1\notin S\) and \({\mathrm n}(S)={\mathrm g}(S)-{\mathrm i}(S).\) As an application of these characterizations, we will provide several algorithms for calculating all the \(\mathrm{PTT}\)-semigroups with a given Frobenius number.

Clustering a signed graph means partitioning the vertices into clusters so that every positive edge, and no negative edge, is within a cluster. The obstruction to clustering is circles with exactly one negative edge (“weakly negative circles’’). The correlation clustering problem is to cluster with the minimum number \(Q\) of edges that violate the clustering rule. A lower bound is \(w\), the maximum number of edge-disjoint weakly negative circles. If every two such circles are edge disjoint, then \(Q=w\). We characterize the signed graphs in which no two weakly negative circles share any edges. A corollary is a straightforward recognition algorithm for such signed graphs. An unsolved problem is to characterize the signed graphs with \(Q=w\).

An equitable vertex \(k\)-coloring of a graph \(G\) is a proper vertex coloring with \(k\) colors such that the size of any two color classes differ by at most one. It was conjectured by Meyer in 1973 that every graph other than the complete graph \(K_n\) or an odd cycle \(C_{2m+1}\) admits an equitable vertex coloring with at most \(\Delta(G)\) colors, where \(\Delta(G)\) is the maximum degree of a vertex in graph \(G\). We show that the conjecture is true for middle graphs of some cycle-based graphs.

A questionnaire is a sequence of multiple choice questions aiming to collect data on a population. We define an abstract questionnaire as an ordered pair \((N,\mathcal{M})\), where \(N\) is a positive integer and \(\mathcal{M}=(m_0,m_1,\ldots,m_{N-1})\) is an \(N\)-tuple of positive integers, with \(m_i\), for \(i \in \mathbb{Z}_N\), as the number of possible answers to question \(i\). An abstract questionnaire may be endowed with a skip-list (which tells us which questions to skip based on the sequence of answers to the earlier questions) and a flag-set (which tells us which sequences of answers are of special interest). An FS-decision tree is a decision tree of an abstract questionnaire that also incorporates the information contained in the skip-list and flag-set. The main objective of this paper is to represent the abstract questionnaire using a directed graph, which we call an FS-decision digraph, that contains the full information of an FS-decision tree, but is in general much more concise. We present an algorithm for constructing a fully reduced FS-decision digraph, and develop the theory that supports it. In addition, we show how to generate all possible orderings of the questions in an abstract questionnaire that respect a given precedence relation.

In \(1973\), Harary and Palmer posed the problem of enumeration of labeled graphs on \(n \geq 1\) unisolated vertices and \(l \geq 0\) edges. In \(1997\), Bender et al. obtained a recurrence relation representing the sequence \(A054548\)(OEIS) of labeled graphs on \(n \geq 0\) unisolated vertices containing \(q \geq \frac{n}{2}\) edges. In \(2020\), Bhavale and Waphare obtained a recurrence relation representing the sequence of fundamental basic blocks on \(n \geq 0\) comparable reducible elements, having nullity \(l \geq \lfloor \frac{n+1}{2} \rfloor\). In this paper, we prove the equivalence of these two sequences. We also provide an edge labeling for a given vertex labeled finite simple graph.

Let \(G\) be a graph with vertex set \(V\) and edge set \(E\) such that every edge \(e\in E\) belongs to at least one copy of a given subgraph \(H\) of \(G\). A bijection \(f:V\cup E\to \{1,2,\dots,|V|+|E|\}\) is called an \(H\)-supermagic labeling if the sum of labels of all vertices and edges of every copy of \(H\) is equal to the same number \(\mu\) and the vertices are labeled with the first \(|V|\) integers. A \(p\)-calendula graph \(Cal_{m,p[n]}\) consists of a cycle \(C_m\) with \(p\) copies of \(C_n\) amalgamated to each edge of \(C_m\). We generalize a previous result by Pradipta and Salman on 1-calendula graphs by providing \(C_n\)-supermagic labelings of \(Cal_{m,p[n]}\) for all \(m,n\geq3, p\geq 1\), and \(m\neq n\). The case of \(m=n, \ p>1\) remains open.

In this note, we present sufficient conditions involving the independence number, the minimum degree, and the maximum degree for some Hamiltonian properties of a graph.