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Electronic Journal of Graph Theory and Applications (EJGTA)
Published by Indonesian Combinatorial Society
ISSN : 23382287     EISSN : -     DOI : -
Core Subject : Engineering,
Electrical & Electronics Engineering
The Electronic Journal of Graph Theory and Applications (EJGTA) is a refereed journal devoted to all areas of modern graph theory together with applications to other fields of mathematics, computer science and other sciences. The journal is published by the Indonesian Combinatorial Society (InaCombS), Graph Theory and Applications (GTA) Research Group - The University of Newcastle - Australia, and Faculty of Mathematics and Natural Sciences - Institut Teknologi Bandung (ITB) Indonesia. Subscription to EJGTA is free. Full-text access to all papers is available for free. All research articles as well as surveys and articles of more general interest are welcome. All papers will be refereed in the normal manner of mathematical journals to maintain the highest standards. This journal is sponsored by CARMA (Computer-Assisted Research Mathematics and its Applications) Priority Research Centre - The University of Newcastle - Australia, and Study Program of Information System- University of Jember - Indonesia.
Arjuna Subject : -
Articles 382 Documents
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A numeral system for the middle-levels graphs Italo J. Dejter
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 9, No 1 (2021): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2021.9.1.13

Abstract

A sequence S of restricted-growth strings unifies the presentation of middle-levels graphs Mk as follows, for 0 < k ∈ Z. Recall Mk is the subgraph in the Hasse diagram of the Boolean lattice 2[2k+1] induced by the k- and (k+1)-levels. The dihedral group D4k+2 acts on Mk via translations mod (2k + 1) and complemented reversals.The first (2k)!/(k!(k+1)!) terms of S stand for the orbits of V(Mk) under such D4k+2-action, via the lexical matching colors 0, 1, ... , k on the k+1 edges at each vertex. So, S is proposed here as a convenient numeral system for the graphs Mk. Color 0 allows to reorder S via an integer sequence that behaves as an idempotent permutation on its first (2k)!/(k!(k+1)!) terms, for each 0 < k ∈ Z. Related properties hold for the remaining colors 1, ... , k.
Some properties of the multiset dimension of graphs Novi H. Bong; Yuqing Lin
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 9, No 1 (2021): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2021.9.1.19

Abstract

The multiset dimension was introduced by Rinovia Simanjuntak et al. as a variation of metric dimension. In this problem, the representation of a vertex v with respect to a resolving set W is expressed as a multiset of distances between v and all vertices in W, including their multiplicities. The multiset dimension is defined to be the minimum cardinality of the resolving set. Clearly, this is at least the metric dimension of a graph. In this paper, we study the properties of the multiset dimension of graphs. 
Total weight choosability for Halin graphs Yu-Chang Liang; Tsai-Lien Wong; Xuding Zhu
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 9, No 1 (2021): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2021.9.1.2

Abstract

A proper total weighting of a graph G is a mapping φ which assigns to each vertex and each edge of G a real number as its weight so that for any edge uv of G, Σe ∈ E(v) φ(e)+φ(v) ≠ Σe ∈ E(u)φ(e)+φ(u). A (k,k')-list assignment of G is a mapping L which assigns to each vertex v a set L(v) of k permissible weights and to each edge e a set L(e) of k' permissible weights. An L-total weighting is a total weighting φ with φ(z) ∈ L(z) for each z ∈ V(G) ∪ E(G). A graph G is called  (k,k')-choosable if for every (k,k')-list assignment L of G, there exists a proper L-total weighting.   As a strenghtening of the well-known 1-2-3 conjecture,  it was conjectured in [Wong and Zhu, Total weight choosability of graphs, J. Graph Theory 66 (2011), 198-212] that every graph without  isolated edge is (1,3)-choosable.  It is easy to verified this conjecture for trees, however, to prove it for wheels seemed to be quite non-trivial.  In this paper, we develop some tools  and techniques  which enable us to prove this conjecture for generalized  Halin graphs.
Multi-switch: A tool for finding potential edge-disjoint 1-factors Tyler Seacrest
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 9, No 1 (2021): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2021.9.1.8

Abstract

Let n be even,  let π = (d1, ... , dn) be a graphic degree sequence, and let π - k = (d1-k, ... , dn-k) also be graphic.  Kundu proved that π has a realization G containing a k-factor, or k-regular graph.  Another way to state the conclusion of Kundu's theorem is that π potentially contains a k-factor. Busch, Ferrara, Hartke, Jacobsen, Kaul, and West conjectured that more was true: π potentially contains k edge-disjoint 1-factors.  Along these lines, they proved π would potentially contain edge-disjoint copies of a (k-2)-factor and two 1-factors. We follow the methods of Busch et al. but introduce a new tool which we call a multi-switch.  Using this new idea, we prove that π potentially has edge-disjoint copies of a (k-4)-factor and four 1-factors. We also prove that π potentially has (⌊k/2⌋+2) edge-disjoint 1-factors, but in this case cannot prove the existence of a large regular graph.
Harmonious graphs from α-trees Christian Barrientos; Sarah Minion
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 9, No 2 (2021): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2021.9.2.9

Abstract

Two of the most studied graph labelings are the types of harmonious and graceful. A harmonious labeling of a graph of size m and order n, is an injective assignment of nonnegative integers smaller than m, such that the weights of the edges, which are defined as the sum of the labels of the end-vertices, are distinct consecutive integers after reducing modulo m. When n = m + 1, exactly two vertices of the graph have the same label. An α-labeling of a tree of size m is a bijective assignment of nonnegative integers, not larger than m, such that the labels on one stable set are smaller than the labels on the other stable set, and the weights of the edges, which are defined as the absolute difference of the labels of the end-vertices, are all distinct; this is the most restrictive type of graceful labeling. Even when these labelings are significantly different in their definitions of the weight, for certain kinds of graphs, there is a deep connection between harmonious and α-labelings. We present new families of harmoniously labeled graphs built on α-labeled trees. Among these new results there are three families of trees, the kth power of the path Pn, the join of a graph G and tK1 where G is a graph that admits a more restrictive type of harmonious labeling and its order is different of its size by at most one unit. We also prove the existence of two families of disconnected harmonius graphs: Kn, m ∪ K1, m − 1 and G ∪ T, where G is a unicyclic graph and T is a tree built with α-trees. In addition, we show that almost all trees admit a harmonious labeling.
On hamiltonicity of 1-tough triangle-free graphs Wei Zheng; Hajo Broersma; Ligong Wang
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 9, No 2 (2021): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2021.9.2.15

Abstract

Let ω(G) denote the number of components of a graph G. A connected graph G is said to be 1-tough if ω(G − X)≤|X| for all X ⊆ V(G) with ω(G − X)>1. It is well-known that every hamiltonian graph is 1-tough, but that the reverse statement is not true in general, and even not for triangle-free graphs. We present two classes of triangle-free graphs for which the reverse statement holds, i.e., for which hamiltonicity and 1-toughness are equivalent. Our two main results give partial answers to two conjectures due to Nikoghosyan.
Embedding complete multi-partite graphs into Cartesian product of paths and cycles R. Sundara Rajan; A. Arul Shantrinal; T.M. Rajalaxmi; Jianxi Fan; Weibei Fan
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 9, No 2 (2021): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2021.9.2.21

Abstract

Graph embedding is a powerful method in parallel computing that maps a guest network G into a host network H. The performance of an embedding can be evaluated by certain parameters, such as the dilation, the edge congestion, and the wirelength. In this manuscript, we obtain the wirelength (exact and minimum) of embedding complete multi-partite graphs into Cartesian product of paths and/or cycles, which include n-cube, n-dimensional mesh (grid), n-dimensional cylinder, and n-dimensional torus, etc., as the subfamilies.
On the k-rainbow domination in graphs with bounded tree-width M. Alambardar Meybodi; M. R. Hooshmandasl; P. Sharifani; Ali Shakiba
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 9, No 2 (2021): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2021.9.2.4

Abstract

Given a positive integer k and a graph G = (V, E), a function f from V to the power set of Ik is called a k-rainbow function if for each vertex v ∈ V, f(v)=∅ implies ∪u ∈ N(v)f(u)=Ik where N(v) is the set of all neighbors of vertex v and Ik = {1, …, k}. Finding a k-rainbow function of minimum weight of ∑v ∈ V|f(v)|, which is called the k-rainbow domination problem, is known to be NP-complete for arbitrary graphs and values of k. In this paper, we propose a dynamic programming algorithm to solve the k-rainbow domination problem for graphs with bounded tree-width tw in ????((2k + 1+1)twn) time, where G has n vertices. Moreover, we also show that the same approach is applicable to solve the weighted k-rainbow domination problem with the same complexity. Therefore, both problems of k-rainbow and weighted k-rainbow domination belong to the class FPT, or fixed parameter tractable, with respect to tree-width. In addition to formally showing the correctness of our algorithms, we also implemented these algorithms to illustrate some examples.
All missing Ramsey numbers for trees versus the four-page book Roland Lortz; Ingrid Mengersen
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 9, No 2 (2021): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2021.9.2.10

Abstract

For the Ramsey number r(Tn, Bm), where Tn denotes a tree of order n and Bm denotes the m-page book K2 + bar(K)m, it is known that r(Tn, Bm)=2n − 1 if n ≥ 3m − 3. In case of n < 3m − 3, r(Tn, Bm) has not been completely evaluated except for m ≤ 3. Here we determine the missing values of r(Tn, B4). Our results close one gap in the table of the Ramsey numbers r(Tn, G) for all trees Tn and all connected graphs G of order six.
On strict-double-bound numbers of graphs and cut sets Kazutaka Ikeda; Kenjiro Ogawa; Satoshi Tagusari; Shin-ichiro Tashiro; Morimasa Tsuchiya
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 9, No 2 (2021): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2021.9.2.16

Abstract

For a poset P=(X,≤P), the strict-double-bound graph of P is the graph sDB(P) on V(sDB(P))=X for which vertices u and v of sDB(P) are adjacent if and only if u ≠ v and there exist elements x,y ∈ X distinct from u and v such that x ≤P u ≤P y and x ≤P v ≤P y. The strict-double-bound number ζ(G) of a graph G is defined as min{ n ; sDB(P) ≅ G ∪ Ǩn {for  some poset P}. We obtain an upper bound of strict-double-bound numbers of graphs with a cut-set generating a complete subgraph. We also estimate upper bounds of strict-double-bound numbers of chordal graphs.

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