Publisher: Indonesian Combinatorial Society (InaCombS)   (Total: 1 journals)   [Sort by number of followers]

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Indonesian J. of Combinatorics     Open Access  
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Indonesian Journal of Combinatorics
Number of Followers: 0  

  This is an Open Access Journal Open Access journal
ISSN (Online) 2541-2205
Published by Indonesian Combinatorial Society (InaCombS) Homepage  [1 journal]
  • The complete short proof of the Berge conjecture

    • Authors: Ikorong Anouk
      Pages: 1 - 41
      Abstract: We say that a graph B is berge if every graph B' ∈ {B,B̄} does not contain an induced cycle of odd length ≥ 5 [B̄ is the complementary graph of B}.A graph G is perfect if every induced subgraph G' of G satisfies χ(G')=ω(G'), where χ(G') is the chromatic number of G' and ω(G') is the clique number of G'. The Berge conjecture states that a graph H is perfect if and only if H is berge. Indeed, the Berge problem (or the difficult part of the Berge conjecture) consists to show that χ(B)=ω(B) for every berge graph B. In this paper, we give the direct short proof of the Berge conjecture by reducing the Berge problem into a simple equation of three unknowns and by using trivial complex calculus coupled with elementary computation and a trivial reformulation of that problem via the reasoning by reduction to absurd [we recall that the Berge conjecture was first proved by Chudnovsky, Robertson, Seymour and Thomas in a paper of at least 143 pages long. That being said, the new proof given in this paper is far more easy and more short].Our work in this paper is original and is completely different from all strong investigations made by Chudnovsky, Robertson, Seymour and Thomas in their manuscript of at least 143 pages long.
      PubDate: 2022-06-27
      DOI: 10.19184/ijc.2022.6.1.1
      Issue No: Vol. 6, No. 1 (2022)
       
  • Prime ideal graphs of commutative rings

    • Authors: Haval Mohammed Salih, Asaad A. Jund
      Pages: 42 - 49
      Abstract: Let R be a finite commutative ring with identity and P be a prime ideal of R. The vertex set is R - {0} and two distinct vertices are adjacent if their product in P. This graph is called the prime ideal graph of R and denoted by ΓP. The relationship among prime ideal, zero-divisor, nilpotent and unit graphs are studied. Also, we show that ΓP is simple connected graph with diameter less than or equal to two and both the clique number and the chromatic number of the graph are equal. Furthermore, it has girth 3 if it contains a cycle. In addition, we compute the number of edges of this graph and investigate some properties of ΓP.
      PubDate: 2022-06-27
      DOI: 10.19184/ijc.2022.6.1.2
      Issue No: Vol. 6, No. 1 (2022)
       
  • The local metric dimension of split and unicyclic graphs

    • Authors: Dinny Fitriani, Anisa Rarasati, Suhadi Wido Saputro, Edy Tri Baskoro
      Pages: 50 - 57
      Abstract: A set W is called a local resolving set of G if the distance of u and v to some elements of W are distinct for every two adjacent vertices u and v in G.  The local metric dimension of G is the minimum cardinality of a local resolving set of G.  A connected graph G is called a split graph if V(G) can be partitioned into two subsets V1 and V2 where an induced subgraph of G by V1 and V2 is a complete graph and an independent set, respectively.  We also consider a graph, namely the unicyclic graph which is a connected graph containing exactly one cycle.  In this paper, we provide a general sharp bounds of local metric dimension of split graph.  We also determine an exact value of local metric dimension of any unicyclic graphs.
      PubDate: 2022-06-28
      DOI: 10.19184/ijc.2022.6.1.3
      Issue No: Vol. 6, No. 1 (2022)
       
  • On graphs with α- and b-edge consecutive edge magic
           labelings

    • Authors: Christian Barrientos
      Pages: 58 - 65
      Abstract: Among the most studied graph labelings we have the varieties called alpha and edge-magic. Even when their definitions seem completely different, these labelings are related. A graceful labeling of a bipartite graph is called an α-labeling if the smaller labels are assigned to vertices of the same stable set. An edge-magic labeling of a graph of size n is said to be b-edge consecutive when its edges are labeled with the integers b+1, b+2, ..., b+n, for some 0 ≤ b ≤ n. In this work, we prove the existence of several b edge-magic labelings for any graph of order m and size m-1 that admits an α-labeling. In addition, we determine the exact value of b induced by the α-labeling, as well as for its reverse, complementary, and reverse complementary labelings.
      PubDate: 2022-06-30
      DOI: 10.19184/ijc.2022.6.1.4
      Issue No: Vol. 6, No. 1 (2022)
       
  • Eigenvalues of Antiadjacency Matrix of Cayley Graph of Z_n

    • Authors: Juan Daniel, Kiki Ariyanti Sugeng, Nora Hariadi
      Pages: 66 - 76
      Abstract: In this paper, we give a relation between the eigenvalues of the antiadjacency matrix of Cay(Z_n, S) and the eigenvalues of the antiadjacency matrix of Cay(Z_n, (Z_n−{0})−S), as well as the eigenvalues of the adjacency matrix of Cay(Z_n, S). Then, we give the characterization of connection set S where the eigenvalues of the antiadjacency matrix of Cay(Z_n, S) are all integers.
      PubDate: 2022-06-30
      DOI: 10.19184/ijc.2022.6.1.5
      Issue No: Vol. 6, No. 1 (2022)
       
 
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