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Decompositions of some classes of regular graphs into cycles and paths of length eight

Cecily Sahai. C, Sampath Kumar. S, Arputha Jose. T

Abstract

Let $C_{k}$ (resp. $P_{k}$) denote the cycle (resp. path) of length $k$. In this paper, we examine the necessary and sufficient conditions for the existence of a $(8; p, q)$-decomposition of tensor product and wreath product of complete graphs.

Decompositions of some classes of regular graphs into cycles and paths of length eight

Abstract

Let (resp. ) denote the cycle (resp. path) of length . In this paper, we examine the necessary and sufficient conditions for the existence of a -decomposition of tensor product and wreath product of complete graphs.
Paper Structure (4 sections, 33 theorems)

This paper contains 4 sections, 33 theorems.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. $(8; p, q)$-decomposition of $(K_{n} \times K_{m})(\lambda)$
  4. $(8; p, q)$-decomposition of $K_{m} \circ \overline{K}_{n}(\lambda)$

Key Result

Theorem 1

A necessary and sufficient condition for the existence of decomposition of a complete multigraph $K_{n}(\lambda)$ into edge disjoint simple paths of length $m$ is $\lambda n(n-1) \equiv 0($mod $2m)$ and $n \geq m+1$

Theorems & Definitions (34)

  • Theorem 1: DCMSPNHD1983
  • Theorem 2: DT2009
  • Theorem 3: NDP1985
  • Theorem 4: CDCM2011
  • Theorem 5: DCBMCS2015
  • Remark 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 24 more