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Even Star Decomposition of Complete Bipartite Graphs
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<p><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;">A decomposition (</span><span><span style="font-family: 宋体; font-size: medium;">G</span></span><sub><span style="font-family: 宋体; font-size: small;">1</span></sub><span style="font-family: 宋体; font-size: medium;">, </span><span><span style="font-family: 宋体; font-size: medium;">G</span></span><sub><span style="font-family: 宋体; font-size: small;">2</span></sub><span style="font-family: 宋体; font-size: medium;">, </span><span><span style="font-family: 宋体; font-size: medium;">G</span></span><sub><span style="font-family: 宋体; font-size: small;">3</span></sub><span style="font-family: 宋体; font-size: medium;">,</span></span><span><span style="font-size: medium;">… </span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;">, </span><span><span style="font-family: 宋体; font-size: medium;">G</span></span><sub><span style="font-family: 宋体; font-size: small;">n</span></sub><span style="font-family: 宋体; font-size: medium;">) of a graph G is an Arithmetic Decomposition(AD) if |</span><span><span style="font-family: 宋体; font-size: medium;">E</span></span><span style="font-family: 宋体; font-size: medium;">(</span><span><span style="font-family: 宋体; font-size: medium;">G</span></span><sub><span style="font-family: 宋体; font-size: small;">i</span></sub><span style="font-family: 宋体; font-size: medium;">)| = a + (i – 1)d for all i = </span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;">1, 2,</span></span><span><span style="font-size: medium;">… </span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;">, n and a, d</span></span><span><span style="font-size: medium;">∈</span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;">Z</span><sup><span style="font-family: 宋体; font-size: small;">+</span></sup><span style="font-family: 宋体; font-size: medium;">. Clearly q = </span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;">n/2</span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;"> [2a + (n – 1)d]. The AD is a CMD if a = 1 and d = 1. In this paper we introduced the new concept Even Decomposition of graphs. If a = 2 and d = 2 in AD, then q = n(n + 1). That is, the number of edges of G </span><span><span style="font-family: 宋体; font-size: medium;">is the sum of first n even numbers 2, 4, 6,</span></span></span><span><span style="font-size: medium;">… </span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;">, 2n. Thus we call the AD with a =</span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;"> 2 and</span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;"> d = 2 as Even Decomposition. Since the number of edges of each subgraph of G is even, we denote the Even Decomposition as (</span><span><span style="font-family: 宋体; font-size: medium;">G</span><sub><span style="font-family: 宋体; font-size: small;">2</span></sub></span><span style="font-family: 宋体; font-size: medium;">, </span><span><span style="font-family: 宋体; font-size: medium;">G</span><sub><span style="font-family: 宋体; font-size: small;">4</span></sub></span><span style="font-family: 宋体; font-size: medium;">,</span></span><span><span style="font-size: medium;">… </span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;">, </span><span><span style="font-family: 宋体; font-size: medium;">G</span></span><sub><span style="font-family: 宋体; font-size: small;">2n</span></sub><span style="font-family: 宋体; font-size: medium;">). </span></span></p><p><span lang="EN-US"><span style="font-family: Calibri; font-size: medium;"> </span></span></p>
Canadian Center of Science and Education
Title: Even Star Decomposition of Complete Bipartite Graphs
Description:
<p><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;">A decomposition (</span><span><span style="font-family: 宋体; font-size: medium;">G</span></span><sub><span style="font-family: 宋体; font-size: small;">1</span></sub><span style="font-family: 宋体; font-size: medium;">, </span><span><span style="font-family: 宋体; font-size: medium;">G</span></span><sub><span style="font-family: 宋体; font-size: small;">2</span></sub><span style="font-family: 宋体; font-size: medium;">, </span><span><span style="font-family: 宋体; font-size: medium;">G</span></span><sub><span style="font-family: 宋体; font-size: small;">3</span></sub><span style="font-family: 宋体; font-size: medium;">,</span></span><span><span style="font-size: medium;">… </span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;">, </span><span><span style="font-family: 宋体; font-size: medium;">G</span></span><sub><span style="font-family: 宋体; font-size: small;">n</span></sub><span style="font-family: 宋体; font-size: medium;">) of a graph G is an Arithmetic Decomposition(AD) if |</span><span><span style="font-family: 宋体; font-size: medium;">E</span></span><span style="font-family: 宋体; font-size: medium;">(</span><span><span style="font-family: 宋体; font-size: medium;">G</span></span><sub><span style="font-family: 宋体; font-size: small;">i</span></sub><span style="font-family: 宋体; font-size: medium;">)| = a + (i – 1)d for all i = </span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;">1, 2,</span></span><span><span style="font-size: medium;">… </span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;">, n and a, d</span></span><span><span style="font-size: medium;">∈</span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;">Z</span><sup><span style="font-family: 宋体; font-size: small;">+</span></sup><span style="font-family: 宋体; font-size: medium;">.
Clearly q = </span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;">n/2</span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;"> [2a + (n – 1)d].
The AD is a CMD if a = 1 and d = 1.
In this paper we introduced the new concept Even Decomposition of graphs.
If a = 2 and d = 2 in AD, then q = n(n + 1).
That is, the number of edges of G </span><span><span style="font-family: 宋体; font-size: medium;">is the sum of first n even numbers 2, 4, 6,</span></span></span><span><span style="font-size: medium;">… </span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;">, 2n.
Thus we call the AD with a =</span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;"> 2 and</span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;"> d = 2 as Even Decomposition.
Since the number of edges of each subgraph of G is even, we denote the Even Decomposition as (</span><span><span style="font-family: 宋体; font-size: medium;">G</span><sub><span style="font-family: 宋体; font-size: small;">2</span></sub></span><span style="font-family: 宋体; font-size: medium;">, </span><span><span style="font-family: 宋体; font-size: medium;">G</span><sub><span style="font-family: 宋体; font-size: small;">4</span></sub></span><span style="font-family: 宋体; font-size: medium;">,</span></span><span><span style="font-size: medium;">… </span></span><span lang="EN-US"><span style="font-family: 宋体; font-size: medium;">, </span><span><span style="font-family: 宋体; font-size: medium;">G</span></span><sub><span style="font-family: 宋体; font-size: small;">2n</span></sub><span style="font-family: 宋体; font-size: medium;">).
</span></span></p><p><span lang="EN-US"><span style="font-family: Calibri; font-size: medium;"> </span></span></p>.
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