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Vol.XVII No.3 Math+CS+Zool JOURNAL OF THE MYANMAR ACADEMY OF ARTS AND SCIENCE Mathematics, Computer Science and Zoology Vol. XVII, No.3, August 2019 Myanmar Academy of Arts and Science Journal of the Myanmar Academy of Arts and Science Vol. XVII, No.3 Contents Section (i) Mathematics Sr. No. Title Page 1 Win Min Aye, *Number of Vertices of Degree 1 Three in Spanning 3-Trees in Square Graphs Section (ii) Computer Studies Sr. No. Title Page 1 Su Myo Swe, *Classification of some Butterflies 13 by Using Rules Induction: Cn2 Algorithm 2 Kyaw Moe Min, Finding Optimal Travelling 39 Salesman Tours Through Myanmar Capital Cities 3 Yee Yee Aung, A Recommender System for 55 Interesting Places in Myanmar by Using Collaborative Filtering Method Journal of the Myanmar Academy of Arts and Science Vol. XVII, No.3 Contents Section (iii) Zoology Sr. No. Title Page 1 Theint Sandar Saing, * Phylogeny of Native Chickens in 67 Central and Western Myanmar 2 Yan Naung Tun, Protozoan Infections in Labeo Rohita 83 (Hamilton, 1822), Piaractus Brachypomus (Cuvier, 1817) and Pangasius Hypopthalmus (Sauvage, 1878) 3 Nay Lin Aung, Effect of Temperature on Hatching and 107 Survival Rates of Nile Tilapia,Oreochromis Niloticus (Linnaeus, 1758) 4 Naing Naing Oo, Larval Development of Black Sea Urchin, 127 Diadema Setosum (Leske, 1778) 5 Cho Cho Thin, Seasonal Variation of Essential and Toxic 143 Metal Contents in Water, Sediment and Some Fishes at Ayeyawady River Segment of Salay Environs 6 Khin July Han, Reproductive Biology of Female Indo- 161 Pacific King Mackerel, Scomberomorus Guttatus (Bloch& Schneider, 1801) from Kyaukphyu Environs, Rakhine State 7 Nwe Nwe Win, Mother - Infant Relationships and Infant 175 Behaviour of Rhesus Macaque (Macaca Mulatta) in Shinma Taung Area, Yesagyo Township, Magway Region 8 Su Su Naing, Assessments on Current Status of Elephant 189 Products Trade in Some Areas of Myanmar 9 Thinzar Oo, Insect Pests and Their Predators on Pomelo 203 Plant in Hmawbi Township 10 Khin War War, Plant-Parasitic Nematodes Associated with 223 Monsoon Rice in Banmaw Environs 11 Yadanar Myo, Species Composition, Abundance and 241 Diversity of Avian Fauna in Pakokku Environs, Magway Region Sr. No. Title Page 12 Myo Sandar Winn, Population, Breeding Status, 257 and Habitats Utilization of Sarus Crane Grus Antigone in Ayeyarwady Region 13 Myint Naung, Speci es Composition of Avifauna 267 in Northern Part of Alaungdaw Kathapa National Park, Sagaing Region 14 Htay Khaing, Species Composition and The 283 Effects of Water Level Fluctuation on Waterbird Population at Inmagyi Wetland in Myinmu Township, Sagaing Region J. Myanmar Acad. Arts Sci. 2019 Vol. XVII. No.3 NUMBER OF VERTICES OF DEGREE THREE IN SPANNING 3-TREES IN SQUARE GRAPHS* Win Min Aye1, Liming Xiong2, Tao Tian3 Abstract In this paper, we show that the square graph of a tree T has a spanning tree of maximum degree at most three and with at most vertices of degree three, where max 0, (txT ( ) 2) 2 xTW 3() there are at least three edge-disjoint paths of length at W3()():T x V T least two that start x} and tx()is the number of edge-disjoint paths with T length at least two that start at a vertex x. Keywords: Square graph; 3-tree; spanning tree. Introduction For graph-theoretic notation not explained in this paper, we refer the reader to J. A. Bondy and U. S. R. Murty, 2008. We consider only simple graph in this paper. Let GVE (,) be a graph with vertex set V and edge set E. A k-tree is a tree with the maximum degree at most k. A graph is called hamiltonian (traceable, respectively) if it has a spanning cycle (path, respectively). Thus a graph is traceable if and only if it has a spanning 2-tree. Therefore, the minimum number of vertices of degree three in a spanning 3-tree F of a graph G shows how closed to be traceable the graph G is. The classic condition for a graph to be traceable is the minimum degree condition, see O. Ore, 1960. It has been extended to consider whether a graph has a spanning k-tree, see S. Win, 1979, in references. It has also been extended to the condition for the existence of a spanning tree with at most k leaves, see H. J. Broersma and H. Tuinstra, 1998. H. J. Broersma and H. Tuinstra gave more structures of the graphs satisfying the condition given by S. Win, 1979; M. Aung and A. Kyaw, 1998, considered the maximum k-tree. V. Neumann-Lara and E. Rivera-Compo, 1991, gave an independence number condition for a graph to have a spanning k-tree with bounded number 1. Dr, Professor, Department of Mathematics, Kalay University. 2. Dr, Professor, School of Mathematics and Statistics, Beijing Institute of Technology. 3. Ph.D Candidate, School of Mathematics and Statistics, Beijing Institute of Technology. * Best Paper Award Winning Paper in Mathematic, (2018) 2 J. Myanmar Acad. Arts Sci. 2019 Vol. XVII. No.3 of vertices with degree k, for k 4. M. Tsugaki, 2009, gave a similar condition for k 3. The square graph of a graph G, denoted by G2 , is the graph with VGVG()()2 in which two vertices are adjacent in G2 if their distance in G is at most two. Thus GG 2 . H . Fleischner, 1974, proved that the square graph of a 2-connected graph is hamiltonian, which was extended by G. Hendry and W. Vogler, 1985. Y. Caro, I. Krasikov and Y. Roditty, 1991, showed that the square graph of a connected graph has a spanning 3-tree. Motivated by the results given above and by the observation that the minimum number of vertices of degree three in a spanning 3-tree F of a graph G2 may measure how closed to be traceable the graph G2 is, Q. Wu, 2016, showed that the square graph of a tree T has a spanning 3-tree F in which every leaf of T has degree one or two and F has at most n p( T ) 3 n 5 max 0,min , vertices of degree three where p(F) is 22 the length of the longest path of F. In the whole paper, we let p(T) be the length of a longest path of a tree T. Theorem 1 Let G be a connected graph of order n. Then G2 has a 3-tree F with at most n p( T ) 3 n 5 min max 0,min , TG 22 vertices of degree three. In this paper, we intend to improve the result above. Firstly, we give the following definitions. Let T be a tree of order n and x a vertex of T. We define txT ()to be the number of edge-disjoint paths with length at least two that start at a vertex x and W3 ()():T x V T there are at least three edge- disjoint paths of length at least two starting at x}. J. Myanmar Acad. Arts Sci. 2019 Vol. XVII. No.3 3 Obviously, tTT()() x d x for any vertex x of T, where dxT ()denote the degree of x in T. For example, for a star TK 1,k , it holds that txT ( ) 0 and d() x k for the center vertex x of K . From the definition of W (T ), one T 1,k 3 may obtain the following observation. Observation 2 Let T be a tree of order n. Then n p( T ) 1 tTT()2() xW3 T (()2) t x . xWW33()() T x T 2 Proof Let P0 be a longest path of T. Then we may obtain T from P0 by adding a path Pi of T such that Pi has a leaf of T, iteratively. In order to increase (tx ( ) 2) at least one, these P (note that, in each step of the xTW () T i 3 proceeding of adding path with a leaf of T, two leaves distance of at least two may be counted once in (tx ( ) 2)should have length at least two. xTW () T 3 Therefore, Observation 2 follows. By Observation 2, in this paper, we continue to give an upper bound for the number of vertices of degree three of spanning 3-tree F in square graph G2 as min max 0,tT ( x ) 2W3 ( T ) 2 TG xTW3 () where T is a spanning tree of G. Hence Theorem 3 Let G be a connected graph of order n. Then G2 has a spanning 3-tree F with at most min max 0,tT ( x ) 2W 3 ( T ) 2 TG xTW3 () 4 J. Myanmar Acad. Arts Sci. 2019 Vol. XVII. No.3 vertices of degree three, where T is a spanning tree of G. Observation 2 shows that the bound in the theorem above improves the one gave in Theorem 1. In the next section, we shall give some auxiliary results, which will be used to proof of Theorem 3 in Section 4. In the last section, we shall show the sharpness of Theorem 3 and Observation 2 and also compare two upper bounds in Theorems 1 and 3, respectively. Preliminaries and Auxiliary Results For SVG ()or E(G), we denote by GS[]the subgraph of G induced by S. For a positive integer s, the graph SK()1,s is obtained from the complete bipartite graph K1,s by subdividing each edge once. The graph G is said to be SK()1,3 -free if it does not contain any induced copy of SK(1,3 ). We use NuG () and duG () to denote the neighbors and the degree of u in G. A leaf or pendant vertex is a vertex of degree one in a graph.
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