https://toc.ui.ac.ir/ Transactions on Combinatorics en daily 1 Tue, 11 Nov 2025 00:00:00 +0330 Tue, 11 Nov 2025 00:00:00 +0330 https://toc.ui.ac.ir/article_29983.html In this paper, it is proved that the identifying code number of the subdivision graph of $G$ is $n$ where $ |V(G)|=n. $This result proves the conjecture posed in [S. Ahmadi, E. Vatandoost and A. Behtoie, Domination number and identifying code number of the subdivision graphs, J. Algebr. Syst., 13 no. 2 (2025) pp. 1--11]. https://toc.ui.ac.ir/article_30045.html We consider the enumeration of circulant matrices over a finite field with $q$ elements, and we provide formulas to compute the number of these circulant matrices for each rank. We also consider the computation of the orders of $q$ in multiplicative groups of integers modulo $n$ that are needed in the enumeration, and we give a reduction method to compute these orders. We then use these results to derive explicit formulas for the enumerations of infinite sequences of circulant matrices of some special types. https://toc.ui.ac.ir/article_30046.html Let $G$ be a graph without isolated vertices. A total dominating set of $G$ is a set $D\subseteq V(G)$ such that every vertex of $G$ is adjacent to some vertex in $D$. A paired dominating set of $G$ is a total dominating set whose induced subgraph has a perfect matching. The total (paired) domination number of $G$ is the minimum cardinality of a total (paired) dominating set of $G$. In this paper, we determine the total and the paired domination numbers of some wheel-related graphs. We also give upper bounds on the total and the paired domination numbers of closed helm graphs and web graphs. Moreover, we determine the paired domination numbers of Jahangir graphs and correct some results on the total domination numbers presented by Mtarneh et al. (Malays. J. Math. Sci., 13(S) (2019) 113--121). https://toc.ui.ac.ir/article_30047.html The Hamiltonian path problem is a well-known problem in graph theory with numerous applications in many fields such as routing, robotics, and parallel processing. In general, this problem is NP-complete for general grid graphs; however, efficient solutions can be found for specific classes of graphs. This paper investigates the Hamiltonian $(s,t)$-path problem in odd-sized $H$-alphabet grid graphs, a sub-class of solid grid graphs. We begin by establishing the conditions under which a Hamiltonian path between two given vertices s and t does not exist. For the cases where a Hamiltonian path exists, we propose an efficient linear-time algorithm to find a Hamiltonian path between the two given vertices.  https://toc.ui.ac.ir/article_30057.html For a graph $G$ with vertices colored either black or white, consider the following rule to change the colors: the color of a vertex which is the only white neighbor of a black vertex, changes from white to black. A proper subset $S$ of the vertex set of $G$ is called a failed zero forcing set if, regardless of how many times this rule is applied to a graph with the initial black vertices $S$, at least one white vertex always remains. The maximum size of such a subset is called the failed zero forcing number of $G$ and is denoted by $F(G)$. In this paper, we look at the failed zero forcing numbers of Grassmann graphs $J_q(n, 2)$ and prove that $F(J_q(n, 2))={n \brack 2}_q - b'_2(q)$, for $n\geq 5$, where $b'_2(q)$ is the maximum number of points in the affine or projective plane of order $q$ such that there is no line that passes through exactly one of these points. Moreover, using maximum arcs in the projective planes, we show that if $q$ is a power of two, then $F(J_q(n, 2))={n \brack 2}_q - (q + 2)$. https://toc.ui.ac.ir/article_30105.html The Colored Points Traveling Salesman Problem (Colored Points TSP) is introduced in this work as a novel variation of the traditional Euclidean Traveling Salesman Problem (TSP) in which the set of points is partitioned into multiple classes, each of which is represented by a distinct color (or label). The goal is to find a minimum cost cycle that visits all the colors so that each color appears only once. This problem finds diverse applications across various fields, including transportation, goods distribution, postal services, inspection, insurance, and banking. By reducing the traditional TSP to it, we can demonstrate that Colored Points TSP is NP-hard. Here, we offer a $\frac{2\pi r}{3}$-approximation algorithm for this problem when the points are placed on a unit grid, where $r$ denotes the radius of the points' smallest color-spanning circle. The algorithm has been implemented, executed on random datasets, and compared against the brute force method. https://toc.ui.ac.ir/article_29813.html Let $t$ and $q$ be positive integers that satisfy $\binom{t+1}{2} \leq q< \binom{t+2}{2}$ and $G$ be a simple and finite graph of size $q$. $G$ is said to be an ascending subgraph decomposition (ASD) graph if $G$ can be decomposed into $t$ subgraphs $H_1, H_2,\ldots,H_t$ without isolated vertices such that $H_i$ is isomorphic to a proper subgraph of $H_{i+1}$, for $1 \leq i \leq t-1$. In this paper, we introduce a new type of antimagic labeling based on the notion of ASD. Let $G$ be an ASD graph and $f:V(G)\cup E(G) \rightarrow \{1,2,\ldots,\lvert V(G)\rvert+\lvert E(G)\rvert\}$ a bijection. The weight of a subgraph $H_i$ $(1\leq i\leq t)$ is $w(H_i)=\sum_{v\in V(H_i)}f(v)+\sum_{e\in E(H_i)}f(e)$. If the weights of all $H_i$s $(1\leq i\leq t)$ form an arithmetic progression with the smallest weight $a$ and common difference $d$, then $f$ is called an $(a,d)$-ASD antimagic labeling and $G$ is an $(a,d)$-ASD antimagic graph. We provide an upper bound for $d$ in an $(a,d)$-ASD antimagic graph. We define and utilize the $(t,\delta)$-ascending antibalanced multisets to label some product graphs, including disjoint union, vertex amalgamation, edge amalgamation, subgraph amalgamation, and extended chain of graphs. https://toc.ui.ac.ir/article_30250.html Let $G$ be a simpple graph. The total $k$-labeling of $G$ is the assignments of a non-negative integer from the set $\{0,2,\ldots,2\lfloor k/2\rfloor\}$ to the vertices and a positive integer from the set $\{1,2,\ldots,k\}$ to the edges of a graph $G$. It is called an edge irregular reflexive $k$-labeling of the graph $G$ if the weights of any two different edges are distinct, where the edge weight is the sum of the label of the edge itself and the labels of its two end vertices. The minimum value $k$ for which the graph $G$ has an edge irregular reflexive $k$-labeling is called the reflexive edge strength of $G$. In this paper, we determine the exact value of the reflexive edge strength for the zigzag graph $Z_{m}^{n}$, where $n \geq 2 $ and $m \geq 3$. https://toc.ui.ac.ir/article_30288.html The star-critical connected Ramsey number $r_c^*(G,H)$, introduced by Moun, Jakhar, and Budden (\textit{TOC}, 2025), is a more deeply studied variant of the connected Ramsey number. For paths versus complete graphs, they obtained the exact values of $r_c^*(P_m,K_n)$ when $m=5$ and when $n=3$. For stars versus complete graphs, they determined the exact values of $r_c^*(K_{1,m},K_n)$ when $m=3$ and when $n=3$. We complete the generalization of both cases by establishing the exact values of $r_c^*(P_m,K_n)$ for $m\ge 6$ and $n\ge 4$, and $r_c^*(K_{1,m},K_n)$ for $m\ge 4$ and $n\ge 4$. The former result disproves a conjecture proposed by Moun et al. (\textit{TOC}, 2025). https://toc.ui.ac.ir/article_30377.html Let $Q$ be a partially ordered set, and let F be an $\ell$-filter in $Q.$ An ideal $P$ in a poset $Q$ with ${\color{red}{P}} \cap F=\emptyset $ is called $F$-prime, if there exists a fixed element $f \in F$ such that whenever $(a,b)^\ell \subseteq P,$ for some $a,b \in Q$ then $(f,a)^\ell \subseteq P$ or $(f,b)^\ell \subseteq P.$ In this paper, we study a topology on the set $Spec_F(Q)$ of all $F$-prime ideals in $Q,$ which is a generalization of the prime spectrum $Spec(Q)$ of a poset $Q.$ We also investigate the relationship between order theoretic properties of $Q$ and topological properties of $Spec_F(Q).$