Papers

Let $H$ be a graph. A graph $G$ is $H$-saturated if $G$ is $H$-free, but adding any edge between two non-adjacent vertices of $G$ yields an $H$-copy as a subgraph. The saturation number $\mathrm{sat}(n, H)$ is the minimum number of edges in an $H$-saturated graph on $n$ vertices. The saturation number for the join of a vertex and a graph $F$, denoted by $K_1\vee F$, has attracted considerable attention. Cameron and Puleo [Discrete Math. 345 (2022), 112867] proved that $\mathrm{sat}(n,K_1 \vee F)\le n-1+\mathrm{sat}(n-1, F)$ for $n > |V(F)|$. A natural question is when the above equality holds. Most existing results impose conditions on $F$ and assume that $F$ has no isolated vertices. Let $K_p^-$ be the graph obtained by deleting one edge from the complete graph $K_p$. In this paper, we investigate the saturation number of $K_1\vee F$ when $F$ contains isolated vertices, and determine the exact value of $\mathrm{sat}(n, K_1\vee F)$ when $F=K^-_{3}\cup sK_1(s\ge 1)$ or $F=K^-_{p-1}\cup K_1(p\ge 5)$. In our results, $\mathrm{sat}(n,K_1 \vee F)= n-1+\mathrm{sat}(n-1, F)$ holds when $F=K^-_{3}\cup sK_1$ for any $s\ge 1$, but fails when $F=K^-_{p-1}\cup K_1$ for $p\ge 5$.

We derive a determinantal expression for the inhomogeneous asymmetric five vertex model in a rectangular geometry with arbitrary boundary conditions at the bottom and top. Standard non-intersecting lattice path, or free fermion, approaches are not applicable and the determinantal form thus is not immediate.

We present an algorithm for computing Hamiltonian cycles that are invariant under a graph automorphism acting on them as a rotation. We also present an application of this algorithm for computing the Hamiltonian compression factor of a graph, that is, the largest order of an automorphism preserving some Hamiltonian cycle and acting on it as a rotation. As an example, we compute the Hamiltonian compression factors of all cubic edge-transitive graphs on up to $10{,}000$ vertices, with the exception of two graphs, which are not Hamiltonian, and $98$ graphs (the smallest having $2304$ vertices) for which only a lower bound for the compression factor is given. As a byproduct, we obtain shortest LCF codes for each of these graphs (except for the two non-Hamiltonian ones; for the $98$ unresolved graphs, the codes obtained are the shortest among those we found).

A sign pattern is a matrix whose entries belong to $\{+, -, 0\}$. Let $\mathcal{P}$ be a symmetric sign pattern and $A$ a real symmetric matrix in its qualitative class. A vertex $v_j$ of the underlying graph of $\mathcal{P}$ is called a $P$-vertex if $ m_{A(j)}(0)-m_A(0)=1,$ where $A(j)$ is the principal submatrix obtained by deleting the $j$-th row and column of $A$, and $m_A(0), ~m_{A(j)}(0)$ denotes the algebraic multiplicity of the eigenvalue $0$ of $A, ~A(j)$, respectively. We say that $\mathcal{P}$ requires full $P$-vertices if every symmetric matrix in its qualitative class has all vertices as $P$-vertices. In this paper, we investigate structural conditions under which symmetric sign patterns require full $P$-vertices. We establish necessary and sufficient conditions for several classes of sign patterns to require full $P$-vertices. In particular, we prove that a tree sign pattern with a $0$-diagonal requires full $P$-vertices if and only if its underlying graph admits a perfect matching. We also derive necessary and sufficient conditions for sign patterns whose underlying graphs contain cycles but no loops to require full $P$-vertices.

In this paper, we explore the theory of higher dimers on band graphs. First, we provide a combinatorial interpretation for the trace of the $q$-deformed higher continued fraction matrices, by showing that with respect to a $q$-weighting on edges, the trace gives the dimer partition function on the set of good higher dimers, which generalizes the notion of good perfect matchings. We also show that the set of good higher dimer covers form a distributive lattice with respect to face flips on square faces. Finally, we attempt to generalize the symmetry result on circular fence posets to the case of good higher dimers, by showing that the dimer partition on a certain family of band graphs are palindromic, in particular, through an approach fitting in the context of dimer theory.

We prove a new combinatorial identity for the determinant of the resistance matrix of a finite graph, which involves counts of spanning trees and forests. This generalizes a result of Graham and Pollak on distances matrices of trees. We make use of Bapat's expression of the resistance matrix determinant as a linear algebraic quantity.

We demonstrate several common techniques for proving linear recurrences from counting Schreier-type sets. These techniques include formula-based arguments, bijective proofs, mathematical induction, the inclusion-exclusion principle, and the characteristic polynomial method. As new contributions, we examine symmetric maximal Schreier sets, Schreier sets that contain a prescribed integer, and Schreier sets that avoid integers belonging to a fixed arithmetic progression. Along the way, we employ useful techniques for identifying meaningful patterns in data and establishing technical identities. The results presented here, together with the diverse proof techniques employed, are expected to serve as a valuable resource for undergraduate researchers interested in this area.

Our aim is to use tropical differential algebra to systematically build a combinatorial basis for the study of the set of (formal) power series solutions to nonlinear algebraic ordinary differential equations (over $\mathbb{C}$) expanded around the point $t_0=0\in\mathbb{C}$, which may also be effectively computed using standard tropical algebra. This paper is divided into two parts. First, given an ordinary tropical differential equation in one differential variable $P=P(y)$ of (differential) order $k$, we study the sets $Sol_{\mathbb{B}[\![t^{\Gamma}]\!],k}(P)\supset \mu(Sol_{\mathbb{B}[\![t^{\Gamma}]\!]}(P)\!)$ of tropical elementary $k$-solutions and minimal tropical solutions, respectively; we show that these two sets bear many similarities. We do this for tropical solutions $y=\varphi(t)\in \mathbb{B}[\![t^{\Gamma}]\!]$ (with coefficients in the boolean semifield $\mathbb{B}$) of $P$ having support in different relevant submonoids $\Gamma$ of $(\mathbb{R},+,0)$. Then, given an ordinary algebraic differential equation $\mathfrak{P}$ (with meromorphic coefficients in one differential variable $\mathfrak{P}=\mathfrak{P}(y)$ and of differential order $k$), we consider the set $S(\mathfrak{P},0):=ord_t(Sol_{\mathbb{C}[\![t^{\mathbb{R}}]\!]}(\mathfrak{P})\!)\subset\mathbb{R}$ of $t$-adic orders of formal Hahn solutions of $\mathfrak{P}$, which is an algebraic object that gives information about the nature of the germs of solutions of $\mathfrak{P}$ at the point $t_0=0\in\mathbb{C}$. We show that this set is contained in the set of $t$-adic orders of Hahn elementary $k$-solutions of its tropicalization $P=trop(\mathfrak{P})$, this is $S(\mathfrak{P},0)\subset ord_t(Sol_{\mathbb{B}[\![t^{\mathbb{R}}]\!],k}(P))$. In most cases, the set $Sol_{\mathbb{B}[\![t^{\mathbb{R}}]\!],k}(P)$ is a finite family of univariate tropical polynomials.

In the last few years, a number of authors have proved divisibility properties satisfied by various functions which count the number of overcubic partition $k$–tuples of weight $n$ for small values of $k$. In this work, we use generating functions to prove some of their results as well as multiple infinite families of new congruences for overcubic partition $k$-tuples which do not yet appear in the literature. In particular, we focus on a new perspective which provides insights as to why these functions are often divisible by powers of 2, and we also prove families of congruences whose moduli are odd. For example, we prove that, for all $m\geq 0$, $\OL{b}_{4}(22m + 11) \equiv 0 \pmod{11}$ and we also prove infinite families such as $\OL{b}_{9l+2}(9m+3) \equiv 0 \pmod{3}$ for all $m,l\geq0$.

In this paper, we investigate which hereditary classes of graphs admit sub-linear (in the number of vertices) bounds on twin-width. By modifying conference graphs, we can show that split, bipartite, and co-bipartite graphs can all have linear twin-width. However, excluding an induced subgraph of each of these three types is equivalent to the class of graphs having bounded VC-dimension, as shown by Bousquet, Lagoutte, Li, Parreau and Thomassé. Graphs of bounded VC-dimension can have unbounded twin-width, but whether it can be linear was an open question. In this paper, we first present a tool for obtaining twin-width bounds in general by contracting a graph based on a partition by distinct neighbourhoods. Then, using this tool, we prove that graphs with bounded VC-dimension have twin-width at most sub-linear. We also obtain a separate, tighter upper bound for the class of interval graphs, as well as a lower bound.

Let $D_d(m)=\operatorname{Cay}((\mathbb Z/m\mathbb Z)^d,\{e_1,\ldots,e_d\})$, with all generators oriented positively. We give a second proof that $D_d(m)$ decomposes into $d$ directed Hamilton cycles for every $d\ge 2$ and every odd $m\ge 3$. The combinatorial core is a fixed-row-sum selection theorem for replicated supports: when each indexed support $A$ is repeated in $m$ identical rows, one can select $\lfloor |A|/2\rfloor$ entries from each row so that every column total is a unit modulo $m$. Applied to the Hamilton factors using a chosen coordinate direction, these selections prescribe the voltages in a cyclic lift that splits the direction into two. In fibre coordinates, the lifted successor is $\widehat h_j(x,z)=(h_j(x),z+\mathbf 1_{\{j\in M(x)\}})$. After one traversal of the base Hamilton cycle, the fibre return is translation by the total carry. Since this carry is a unit modulo $m$, the return is a single $m$-cycle and the lifted factor is Hamilton. The new fibres also preserve the direction-constant block structure required for the next split. Iterating from a directed $m$-cycle with $d$ parallel copies of each arc yields the desired decomposition. The proof strategy was proposed with the assistance of OpenAI GPT-5.5 Pro and formally verified in Lean 4.

An $\ell$-cycle decomposition of a graph $G$ is a set of $\ell$-cycles in $G$ whose edge sets partition the edge set of $G$. The $\ell$-cycle decomposition threshold $\delta_{C_\ell}$ is then the least real number such that any $n$-vertex graph $G$ with minimum degree at least $(\delta_{C_\ell}+o(1))n$ has an $\ell$-cycle decomposition if and only if $\ell$ divides $|E(G)|$ and each vertex of $G$ has even degree. Nash-Williams' famous conjecture on triangle decompositions states, asymptotically, that $\delta_{C_3}=\frac{3}{4}$. A very recent breakthrough result of Delcourt and Postle completely resolved this conjecture, however, Glock, Kühn, and Osthus have posed the problem of determining $\delta_{C_\ell}$ for larger odd values of $\ell$ (the behaviour of $\delta_{C_\ell}$ for even $\ell$ is different and well understood). A natural generalisation of Nash-Williams' conjecture implies that $\delta_{C_\ell}=\frac{\ell}{2\ell-2}$ for all odd $\ell \geq 3$. Here we prove that this conjecture holds for all $\ell \geq 73$.

A graph $G$ is conformable if it admits a proper $(\Delta(G)+1)$-coloring in which, among the $\Delta(G)+1$ color classes including the empty ones, at most $\sum_{v\in V(G)}(\Delta(G)-d_G(v))$ have parity different from that of $|V(G)|$. The complexity of deciding conformability was left open in recent work, and positive results for several graph classes had suggested that the problem might be polynomial-time solvable. We settle the general problem by proving that Conformability is NP-complete. Hardness holds even for connected regular graphs $G$ of odd order with independence number $\alpha(G)=3$ and maximum degree $\Delta(G)\ge |V(G)|/2$. In particular, NP-completeness persists when every color class is forced to have the parity of the order. The reduction starts from perfect triangle packing in graphs of clique number three, regularizes the source graph while preserving the relevant triangle packings, and then takes the complement. In the complement, conformable color classes correspond to odd cliques of the regularized graph; $K_4$-freeness restricts these cliques to singletons or triangles, and the number of available colors forces exactly the required number of disjoint triangles.

In a 2005 paper, Bebeacua et al. investigated the xrays of permutations, and conjectured a correspondence between binary xrays and score sequences of tournaments. In 2014, Brualdi and Fritscher conjectured a possible correspondence between score sequences of $2$-tournaments and (not necessarily binary) xrays of permutations. In this paper, we first introduce the concept of a transitive tournament decomposition of $k$-tournaments, then present a construction by which a permutation is used to build $1$- and $2$-tournaments whose score sequences agree with the xray of the permutation in the manner outlined by Bebeacua et al. and Brualdi and Fritscher. We close with an investigation of xrays with restricted terms, including binary xrays, and show that the recent conjectures by Bebeacua et al. and Brualdi and Fritscher are special cases of a more general statement, which we conjecture and for which we provide supporting evidence.

We introduce a combinatorial structure $(n,W,T)$ encoding the topological type of a curve transverse to a fixed cellular arrangement of curves on a compact real surface, in terms of intersection numbers, Dyck words and rooted trees. We apply this formalism to analyze a natural generalization of Hilbert's 16th problem to arrangements of curves. We obtain a complete classification of arrangements of three lines and a cubic, and a partial classification of arrangements of three lines and a quartic. This is achieved using Bézout-type obstructions, Viro's patchworking and translations, and by developing the Julia library NWT to handle large databases of curves.

This paper is the third part of a series that intends to study the resolving subcategories for gentle algebras over an algebraically closed field $\mathbb{K}$. As in the previous two papers, we continue to focus on gentle trees $(Q,R)$. Via a modified surface model for gentle algebras with finite global dimension, we developed combinatorial, poset, and quiver representation techniques that allow one to calculate all the resolving subcategories of $\mathbb{K}Q/\langle R \rangle$-mod. Furthermore, they enable one to calculate the resolving subcategory generated by any collection of $\mathbb{K}Q/\langle R \rangle$-modules. In this paper, based on those techniques, we give a combinatorial realization of the Auslander–Reiten one-to-one correspondence between resolving subcategories and tilting modules in $\mathbb{K}Q/\langle R \rangle$-mod.

Given a central hyperplane arrangement $\mathcal{A}$ defined over a field of characteristic zero, we construct matrices of linear forms whose local ask zeta functions are recovered by the Igusa local zeta function of the cone over $\mathcal{A}$. Our construction extends a previously established connection between ask zeta function of hypergraphs and the Igusa local zeta function of Boolean arrangements. From a combinatorial standpoint, we introduce the truncated flag Hilbert-Poincaré series of $\mathcal{A}$, obtained as a rank specialisation of the flag Hilbert-Poincaré series of the cone over $\mathcal{A}$. Whenever $\mathcal{A}$ admits good reduction over the finite field $\mathbb{F}_q$, suitable substitutions of the variables of its truncated flag Hilbert-Poincaré series recover the local ask zeta functions associated with $\mathcal{A}$. Such formulae provide a means to study the analytic properties of the ask zeta functions considered, as well as to derive their reduced and topological relatives.

Hasunuma [Graphs Combin. 41:10 (2025)] proved that for $k\ge 2$, there exists a function $f(k)=O(k)$ such that every $k$-connected graph $G$ of order $n\ge f(k)$ with $\delta(G)\ge \frac{n}{2}$ contains a Hamiltonian cycle $H$ such that $G-E(H)$ is $k$-connected. In this paper, we show that for $k\ge 2$, if $G$ is a $k$-connected graph of order $n\ge 6k+6$ with minimum degree at least $\frac{n+1}{2}$, then for any two distinct vertices $u,v\in V(G)$, there exists a Hamiltonian $(u,v)$-path $P$ such that $G-E(P)$ is $k$-connected. Moreover, we further extend this result to $s$ internally disjoint spanning $(u,v)$-paths.

A family of permutations of $[n]$ is called setwise distinguishable if for every permutation in the family there exists a subset of $[n]$ whose image under this permutation differs from its image under any other permutation in the family. We prove that there exists a setwise distinguishable family of $2^{(2-o(1)) \cdot n}$ permutations of $[n]$. The result is optimal up to the $o(1)$ term in the exponent and is achieved through an explicit construction. As an application, we obtain nearly tight conditional lower bounds on the kernelization complexity of graph coloring problems parameterized by the vertex-deletion distance to split graphs. This improves a result of Jansen and Kratsch (Inf. Comput., 2013).

The directed support graph of a discrete-time, time-homogeneous Markov chain naturally defines a regular language whose words are the admissible trajectories from a fixed initial state. Applying the Parikh map associates with each trajectory its occupation vector, recording how many times each state is visited while disregarding the chronological order of the visits. Equivalently, each trajectory determines a monomial whose exponents are its occupation numbers, yielding a natural commutative representation of occupation patterns. For each trajectory length n, we define the occupation ideal generated by the corresponding Parikh monomials. The minimal generators of this ideal are in one-to-one correspondence with the distinct occupation patterns realized at that length, providing an algebraic encoding of the combinatorial structure of admissible trajectories. This construction naturally gives rise to three complementary measures of support complexity describing reachability growth, trajectory growth, and occupation-pattern growth. The resulting framework connects Markov support graphs, regular languages, Parikh images, and monomial ideals, establishing occupation ideals as a new algebraic object associated with Markov support dynamics. It provides a natural interface between stochastic processes, formal language theory, and combinatorial commutative algebra, and suggests new algebraic and geometric approaches to the study of support dynamics. Examples illustrate how occupation patterns reflect branching, recurrence, transience, and local oscillation in the underlying support graph.

Let $K$ be a finite $r$-dimensional simplicial complex with vertex set $V$ of size $n$. We study the largest eigenvalue of the combinatorial $(r-1)$-up Laplacian $L^{\operatorname{up}}_{r-1}(K)$. It is known that \[ \lambda_{\max}\bigl(L^{\operatorname{up}}_{r-1}(K)\bigr)\le n. \] We first give a homological equality criterion for this universal bound, namely, the equality holds if and only if the $r$-dimensional complement $K^c$ of $K$ has a nonzero reduced homology $\widetilde H_{r-1}(K^c,\mathbb{R})$. For $r=1$, this is the classical graph condition that the complement graph is disconnected. Secondly, we prove a sharper upper bound for $\lambda_{\max}(L^{\operatorname{up}}_{r-1}(K))$: \[ \lambda_{\max}(L^{\operatorname{up}}_{r-1}(K)) \le \max_{F\in S_r(K)} \bigl|\bigcup_{E \in \partial F} N_K(E) \bigr| \le n,\] where, for an $(r-1)$-face $E$, $N_K(E)$ denotes the set of vertices $u$ outside $E$ such that the union $E \cup \{u\}$ is an $r$-face of $K$. This is the high-dimensional analog of the graph Laplacian bound. We give an explicit characterization of the equality case, and construct a broad family attaining the bound, namely, the partite semiregular complexes with admissible additions.

For a graph $G=(V,E),$ a set $S \subseteq V$ is called an isolating set of $G$ if the set $V-N[S]$ is independent. The minimum cardinality of an isolating set in $G$ is the isolation number of $G$, denoted by $\iota(G).$ Here we introduce the isolation subdivision number of a graph $G$, denoted by ${\rm sd}_\iota(G)$, as the minimum number of edges of $G$ that must be subdivided, where each edge can be subdivided at most once, in order to obtain a graph with isolation number greater than $\iota(G).$ We show that the new parameter is well defined for any non-trivial graph different from a star and that it can be arbitrarily large. We present the values of this parameter for some elementary classes of graphs and establish some basic properties. We show also that $1\leq {\rm sd}_\iota(T)\leq 4$ for any tree $T$ different from a star and characterize all trees $T$ with ${\rm sd}_\iota(T)=1.$

Automated floor plan generation is an important problem in computational architectural design. The goal is to construct a floor plan from user-defined room numbers and door requirements. The user specifies which rooms must share a door and which rooms must not be adjacent. However, these requirements do not determine the exact placement or shape of the rooms. The task is therefore to arrange the rooms in a single floor plan so that all required door connections are satisfied and no rooms overlap. To address this problem, we propose DPLAN (Door Connectivity to Floor Plan Generation), a graph-based prototype that generates floor plans from door and non-adjacency constraints. The framework operates in three stages. First, the user-defined graph is examined and, if disconnected, additional edges are added to connect its components. Second, a bi-connected plane triangulation is constructed to ensure the existence of a floor plan without overlapping rooms or empty spaces. Third, the triangulated graph is transformed into floor plans. For rectangular floor plans (RFPs), separating triangles are removed by modifying edges without adding new vertices, thereby avoiding the creation of extra rooms. For orthogonal floor plans (OFPs), separating triangles are removed by introducing additional vertices, allowing rectilinear room shapes. By enforcing both door and non-adjacency requirements, the framework generates floor plans that satisfy the given constraints. The method is implemented in Python and includes a prototype for interactive constraint specification and floor plan visualization. Currently, the framework supports rectangular plot boundaries. Future work includes support for non-rectangular plots, dimension-based scaling, and circulation modeling.

This work solves the enumeration problem for weighted bi-colored plane trees with prescribed numbers of black and white vertices, together with prescribed total edge weights at each vertex. An exact closed formula for a particular case is obtained, and a unified algorithmic method for the general case is provided. We then apply this result to two geometric problems. Firstly, we compute the strong Hurwitz number for a special class of branch datum between Riemann spheres with three branched points. This is done by counting the dessins d'enfants that faithfully record such branched covers, and a clear correspondence between dessins and weighted trees in such case. Secondly, we study the geometric moduli space for a special class of extremal Kähler metrics on Riemann sphere (HCMU spheres), with a single conical singularity. We classify and enumerate the connected components of the moduli space with respect to the Gromov-Hausdorff topology. This is based on an efficient representation of these metric surfaces, and a study of Gromov-Hausdorff limits of such surfaces.

We study the multigraded regularity of the complete flag variety under the Plücker embedding. In particular, we prove inductive relationships about the regularity regions, and we provide some inner and outer bounds on the regions.

In 2022 Aliniaeifard, Li, and van Willigenburg defined Schur functions in the algebra of symmetric functions in noncommuting variables (NCSym), answering an open question posed by Rosas and Sagan in 2004. These Schur functions are not monomial positive, since they are defined via a noncommutative analogue of the Jacobi-Trudi determinant. We introduce Hall-Littlewood functions ${\bf P}_{\pi}({\bf x};t)$ indexed by set partitions $\pi$ in noncommuting variables ${\bf x}=({\bf x}_1,{\bf x}_2,\ldots)$, and define Schur functions in noncommuting variables to be ${\bf s}_{\pi}({\bf x})={\bf P}_{\pi}({\bf x};0)$. We prove that the set of Hall-Littlewood functions $\{{\bf P}_{\pi}({\bf x};t)\}$ for all set partitions $\pi$ of $[n]$ forms a $\mathbb{Q}[t]$-basis of NCSym of homogeneous degree $n$, and that this basis is invariant under any permutation acting on set partitions. These Hall-Littlewood functions in NCSym map to classical Hall-Littlewood functions under commutation, up to a scalar factor. We also show that the Hall-Littlewood functions ${\bf P}_{\pi}({\bf x};t)$ naturally refine the lifted Hall-Littlewood functions in NCSym. Specifically, the Schur functions ${\bf s}_{\pi}({\bf x})$ are monomial positive and refine the lifted Schur function introduced by Rosas and Sagan. Moreover, we introduce a star product of two polynomials in NCSym and develop the star-multiplication rule for a lifted and a non-lifted Hall-Littlewood functions in NCSym. This rule is a noncommutative analogue of the product rule for two Hall-Littlewood functions and, in particular, of the Littlewood-Richardson rule. Finally, our approach extends to the algebra of quasisymmetric functions in noncommuting variables (NCQSym) indexed by set compositions.

Sabok asked whether the compact convex set \(S'(X)\) attached to a separable metric space of diameter at most one is always a simplex, and whether \(S'(\mathbb U_1)\) is the Poulsen simplex. We give negative answers. For finite \(X=\{x_1,\ldots,x_m\}\), \(S'(X)\) is affinely homeomorphic to the convex hull of the rows \(r_i=(d(x_i,x_1),\ldots,d(x_i,x_m))\) of the distance matrix; it is a simplex exactly when these rows are affinely independent. The diameter-one four-cycle gives the minimal finite obstruction. For the Urysohn sphere, using the rational Urysohn sphere \(D\) as coordinates, we identify the coordinate model \(S'_D(\mathbb U_1)\) with the Katétov compactum \(K(D)\). Four explicit extreme points \(f_A,g_A,\mathbf 1,\mathbf h\) satisfy \(f_A+g_A=\mathbf 1+\mathbf h\), giving two distinct representing measures for \((3/4)\mathbf 1\). Hence \(S'(\mathbb U_1)\) is not a Choquet simplex.

In this article, we introduce a group linear representation associated to a graph. We study this representation as well as the canonical decomposition of the associated module. A character study is conducted, demonstrating the relevance of this approach to graph theory. We also develop the links that exist between algebra and graphs, particularly in the language of representations.

Given a finite sequence of integers, form its difference triangle by repeatedly taking absolute differences of consecutive entries. We call the sequence Gilbreath if the leftmost entry of every row below the top is 1. The Gilbreath conjecture, which remains open, asserts that every initial segment of the primes is a Gilbreath sequence. This paper studies the local extension problem: given a Gilbreath sequence, which integers can be appended to it while preserving the Gilbreath property? We call the set of such admissible values the valid-extension set of the sequence. A previously proposed characterization in the literature predicts that this set always fills a natural parity interval around the last term. We show that this fails in general: the valid-extension set can have interior holes, with the smallest failure occurring at length 5 for the sequence (2,3,5,9,15). The paper develops a corrected theory of valid extensions. We give an exact criterion for membership in the valid-extension set, an algorithm that computes it, and a sharp condition determining exactly when the set fills the candidate interval. This last condition is an order-sensitive analogue of the classical Brown completeness criterion for subset sums. We also establish endpoint validity and reflection symmetry, determine the exact minimum size of the valid-extension set together with its unique minimizer, exhibit a family whose valid-extension set has exponentially many components, and provide enumeration data through length 11.

This study is the $7^{th}$ part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10}. In this study, we define isomorphic set, isomorphism series, isomorphism digraph $\mathcal{D}$ or isomorphism diagram and isomorphism graph $\mathcal{G}$ of circulant graphs and obtain these corresponding to $C_{16}(R)$, $C_{27}(S)$ and $C_{54}(1,3,17,19)$ and present the isomorphism digraph and the isomorphism graph of $C_{432}(16, 27, 48, 54, 128, 160, 189)$ which has isomorphic circulant graphs of Type-2 w.r.t. $m$ = 2 as well as $m$ = 3. We also show that each pair of circulant graphs $C_{54}(1,3,17,19)$, $C_{54}(5,13,21,23)$; $C_{54}(7, 11, 21, 25)$, $C_{54}(7, 11, 15, 25)$; and $C_{54}(1,3,17,19)$, $C_{54}(7,11,15,25)$ are isomorphic but they are neither of Type-1 nor of Type-2 w.r.t. $m$ = 3. More such circulant graphs are given in the conclusion. We also define diameter of isomorphic set of $C_n(R)$ and isomorphic distance of $C_n(S)$ and $C_n(T)$ and obtained these values for some circulant graphs.