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Properties of Graphs With H-Covering and Prescribed H-Weights

Thesis Info

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Author

Umar, Muhammad Awais

Program

PhD

Institute

Government College University

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2018

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/12928/1/Muhammad_Awais_Umar_Maths_HSR_2018_GCU_Lahore_03.04.2018.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727038250

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Let vertex and edge sets of graph G are denoted by V (G) and E(G), respectively. An edge-covering of G is a family of di erent subgraphs H1;H2; : : : ;Hk such that each edge of E(G) belongs to at least one of the subgraphs Hj , 1 j k. Then it is said that G admits an (H1;H2; : : : ;Hk)-(edge)covering. If every Hj is isomorphic to a given graph H, then G admits an H-covering. For axed graph H, a total labeling : V (G) [ E(G) ! f1; 2; : : : ; jV (G)j + jE(G)jg is said to be H-magic if all subgraphs of G isomorphic to H have the same weight. One can ask for di erent properties of a total labeling. The total labeling is said to be antimagic if the weights of subgraphs isomorphic to H are pairwise distinct. Further restriction on the weights of subgraphs provides (a; d)-H-antimagic labelings where the weights of subgraphs form an arithmetic progression with di erence d and rst element a. If graph G is a 2-connected plane graph then the H-antimagic labeling is equiva- lent to d-antimagic labeling of type (1; 1; 0), where weights of all faces form an arith- metic sequence having a common di erence d and the weight of a face under a labeling of type (1; 1; 0) is the sum of labels carried by the edges and vertices on its boundary. In therst part of the thesis we will study the notions, notations and de nitions about graphs and labeling of graphs. In the second part of the thesis, we have three chapters on newly obtained results. In the chapters, we examine the existence of Hk 2 -supermagic labelings for graphs Gk 2 obtained from two isomorphic graphs G and G0 by joining every couple of corre- sponding vertices v 2 V (G) and v0 2 V (G0) by a path of length k + 1. We show that graphs Gk(w), obtained from a graph G by joining all vertices in G to a vertex w by paths of length k + 1, keep super H-antimagic properties of the graph G. We also examine the existence of the (H G2)-supermagic labelings of Cartesian product G1 G2, where G1 admits an H-covering and G2 is a graph of even order. Addition- ally, we show that if a graph G admits a (super) (a; 1)-tree-antimagic labeling then the disjoint union of multiple copies of the graph G keeps the same property.
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