Indonesian Journal of Combinatorics (IJC) publishes current research articles in any area of combinatorics and graph theory such as graph labelings, optimal network problems, metric dimension, graph coloring, rainbow connection and other related topics. IJC is published by the Indonesian Combinatorial Society (InaCombS), CGANT Research Group Universitas Jember (UNEJ), and Department of Mathematics Universitas Indonesia (UI). All papers will be refereed in the normal manner of mathematical journals to maintain the high standards. IJC is an open access journal. Full-text access to all papers is available for free. This journal is sponsored by Indonesian Mathematical Society (IndoMS) and Faculty of Mathematics and Natural Sciences - Institut Teknologi Bandung (ITB) Indonesia. Indonesian Journal of Combinatorics (IJC) has been accredited by National Journal Accreditation (ARJUNA) Managed by Ministry of Research, Technology, and Higher Education, Republic Indonesia with Second Grade (Sinta 2) since 2017 to 2022 according to the decree No. 10/E/KPT/2019 and re-acredited by Ministry of Education, Culture, Research and Technology with the same Grade (Sinta 2) until 2026 according the decree No. 204/E/KPT/2022.
Indonesian Combinatorial Society (InaCombS)
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In this paper, we find the number of non-trivial tripotent elements for some finite rings, namely ℤn, H(𝔽q), and 𝔽q Cn. For this purpose, we find the general formula of them. Furthermore, we introduce the tri-potent graph of a finite ring R, denoted by Tri(R), where two distinct vertices x and y in R with a<b are adjacent if and only if a − b ∈ Tri(R). It is shown that the tri-potent graph is a bi-regular connected with girth at most 4. Also, the tri potent graph of ℤn is bipartite graph for some n.
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Partition dimension of graphs with two bridges on rose graphs
The partition dimension of a graph G, denoted by pd(G) is a generalization of the metric dimension, in which the distinction between vertices is no longer based on a specific set of vertices, but rather on a partition of the vertex set of the graph. A partition is called a resolving partition if every vertex in the graph has a distinct distance vector representation with respect to each subset in the partition. The minimum cardinality of such a resolving partition is called the partition dimension of the graph. This study focuses on the partition dimension of double bridge graphs constructed from a pair of rose graphs. It is shown that the partition dimension of the double bridge graph obtained from two rose graphs connected by two bridge edges is 4
Totally antimagic total labeling of helm and gear graphs
A total labeling of a graph G is a bijection from the union of the vertex set and the edge set of G to the set {1,2,...,|V(G)|+|E(G)|}. Under a total labeling, the vertex-weight of a vertex is defined as the sum of its label and the labels of all edges incident to it. Similarly, the edge-weight of an edge is the sum of its label and the labels of its two end vertices. A total labeling is said to be edge-antimagic total if all the edge-weights are pairwise distinct, and vertex-antimagic total if all the vertex-weights are pairwise distinct. If a total labeling is edge-antimagic total and vertex-antimagic total at the same time, then it is called a totally antimagic total labeling. A graph that admits totally antimagic total labeling is called a totally antimagic total graph. In this paper, we show that helm graphs Hn and gear graphs Gn are totally antimagic total graphs.
A note on line and total directed superhypergraphs, line bidirected graphs, line multidirected graphs, and related structures
Hypergraphs extend classical graphs by allowing hyperedges to connect any nonempty subset of vertices, thereby capturing complex group-level relationships. Superhypergraphs advance this framework by introducing recursively nested powerset layers, enabling the representation of hierarchical and self-referential links among hyperedges. A line graph encodes the adjacencies between edges of an original graph by transforming each edge into a vertex and connecting two vertices if their corresponding edges share a common endpoint. A total graph incorporates both the vertices and edges of the original graph as its own vertices, with edges representing adjacency or incidence between these entities. Various extensions of these graph concepts exist that incorporate directional information, such as Directed Graphs, Bidirected Graphs, and Multidirected Graphs. In this paper, we investigate the notions of line graphs and total graphs within the settings of Directed HyperGraphs, Directed SuperHyperGraphs, Bidirected Graphs, and Multidirected Graphs.
On rainbow antimagic coloring and local edge antimagic coloring of graphs
Let G be a connected graph of order n. Let f: V(G) → {1,2,...,n} be a bijection and for every uv ∈ E(G) consider wf(uv) = f(u) + f(v) as a coloring of the edge. For a pair of vertices u and v, they are connected by a rainbow path if there exists a path from u to v such that the edges have pairwise distinct colors from wf. The bijection f: V(G) → {1,2,...,n} is a rainbow antimagic coloring if for every two vertices there exists a rainbow path. Meanwhile, that bijection f: V(G) → {1,2,...,n} is local edge antimagic coloring if every two adjacent edges have distinct weights. The rainbow antimagic connection number rac(G) and local edge antimagic chromatic number χ'lea(G) is the minimum number of distinct edge weights over all rainbow antimagic coloring and local edge antimagic coloring, respectively. We investigate the relationship between rac(G) and χ'lea(G). We prove χ'lea(G) ≤ rac(G) for all graphs and provide conditions for equality. Graphs with diameter at most 2 or satisfying rac(G) = Δ(G) achieve equality. We construct a family Ad with arbitrarily large diameter d where Δ(Ad) = χ'lea(Ad) = rac(Ad), and a family Hn = Kn,n - nK2 of diameter 3 where χ'lea(Hn) = rac(Hn) = 2n-3 > Δ(Hn). These results present hints to the full characterization of graphs G with χ'lea(G) = rac(G).
Prime and odd prime labelings of several graph classes
A prime labeling of a graph is an assignment of distinct integers to its vertices such that any two adjacent vertices are labeled with numbers that are relatively prime. An odd prime labeling is a variant in which the labels are restricted to odd integers while still preserving the coprimality condition. Motivated by the odd prime graph conjecture, which asserts that every prime graph is odd prime, we study prime and odd prime labelings on various classes of graphs. We construct explicit labeling functions and prove that comb graphs, disjoint unions of comb graphs, triangular book graphs, and modified triangular book graphs K1,1,n ⊙ 2Sm admit both prime and odd prime labelings. We further establish an odd prime labeling for torch graphs, extending a known prime labeling result for this class. Consequently, all graph families considered in this paper satisfy the odd prime graph conjecture. These results expand the collection of graphs known to admit odd prime labelings and provide additional evidence supporting the conjecture.