A prime labeling of a simple finite graph \(G\) is a labeling of the vertices with distinct integers from \(\{1,2, \dots, |G|\}\) such that the labels of any two adjacent vertices are coprime. If \(G\) admits such a labeling, we call \(G\) a prime graph. If the set \(\{1,2, \dots, |G|\}\) is replaced by \(\{k,k+1, \dots, k+|G|-1\}\) in the previous definition, then we call it \(k\)-prime labeling, where \(k\) is a positive integer. Since some graph might be 2-prime labeling but not 3-prime labeling, to avoid ambiguity, we define uniform prime labeling for graphs. Focused on trees, using some very classical number theory results, we show that any star is uniform prime if and only if it has at most 16 vertices. We also prove any tree of order \(n\) with diameter \(n-1\), \(n-2\), \(n-3\), and \(n-4\) are uniform prime. Based on these results and several trees with diameter \(n-5\), we show that any tree with at most 8 vertices (There are 1+1+1+2+3+6+11+23=48 such non-isomorphic trees) are uniform prime. We also verified the uniform primality of several periodically constructed trees. We post some open questions and conjectures which should be essential and interesting by the end of Section 4. For example, we conjecture all the trees with order up to 16 are uniform prime and ask if there exists a tree other than star is not uniform prime.

Let \(G(V,E)\) be a finite simple graph. Denote by \(|V|\) and \(|E|\) the cardinality of the set \(V\) and \(E\), respectively. An \(\alpha\)-labeling \(f\) is an injective function \({f}:V\rightarrow \{0,1,2,…, |E|\}\) such that \(\{|f(u)-f(v)|:uv\in E\}=\{1,2,…,|E|\}\) and for a constant \(\lambda\), \(f(u)\le \lambda <f(v)\) for every \(uv\in E\). A graph that has an \(\alpha\)-labeling is called \(\alpha\)-graph. An open chain graph, or shortly a chain graph, is a graph with blocks \({B_1,B_2,…,B_n}\) such that for every \(i\), \(B_i\) and \(B_{i+1}\) have a common vertex, in such a way that the block-cut-vertex graph is a path. A chain graph having \({n}\) blocks \({B_1,B_2,…,B_n}\) is denoted by \([B_{1},B_{2},…,B_{n}]\). Let \({c_i}\) be the common vertex of \({B_i}\) and \({B_{i+1}}\), in \([{B_1,B_2,…,B_n}]\), \(1\le i \le n-1\). Consider a vertex of \({B_1}\), \({c_0 \ne c_1}\), and a vertex of \({B_n}\), \({c_n \ne c_{n-1}}\). Assume that \(B\) is another block graph and \(x\) and \(y\) are two different vertices in \(B\). The graph which is constructed by identifying \(c_0\) with \(x\) and \(c_n\) with \(y\) is called closed chain graph, and is denoted by \([{B_1,B_2,…,B_n,B}]_c\). By using pattern recognition and axiomatic deductive method we get results that some chain graphs with blocks of complete bipartite graphs are \(\alpha\)-labeling.

Threshold graphs are graphs whose node set can be partitioned into a clique and an independent set, with the additional property that for each pair of nodes, one’s neighborhood is a subset of the other’s neighborhood. Threshold graphs have been well-studied in graph theory, but not much is known about multigraphs that are underlying threshold. Proper threshold graphs are those in which all nodes in the independent set have the same degree. In this paper, we present a formula for the eigenvalues of a particular class of multigraphs that are underlying proper threshold.

Recently, the notion of a Weyl substructure in a spherical building was introduced type by type. In this paper we provide a uniform (axiomatic) definition across all types. In particular, this provides a new characterisation of the Ree-Tits octagons. We then show that uniclass automorphisms of spherical buildings are uniformly characterised by their fix structure. For type preserving automorphisms, this follows from earlier work, and so the focus here is on dualities. In particular, it follows that a duality pointwise fixing a Weyl substructure is automatically a polarity. This characterises all polarities in self-dual spherical buildings where opposition acts trivially on the types.

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 \(Q=w\).