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On generating $k$-factorable graphic sequences with connected (resp.no connected) $k$-factors

Asish Mukhopadhyay, Daniel John, Lucas Sarweh

Abstract

In this note, we consider the problem of generating $k$-factorable graphic sequences with connected (resp. no connected) $k$-factors.

On generating $k$-factorable graphic sequences with connected (resp.no connected) $k$-factors

Abstract

In this note, we consider the problem of generating -factorable graphic sequences with connected (resp. no connected) -factors.

Paper Structure

This paper contains 12 sections, 5 theorems, 8 equations, 16 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Generating $k$-factorable graphic sequences with connected $k$-factors
  3. Further considerations
  4. Generating $k$-factorable graphic sequences with no connected $k$-factors
  5. Case $k = 2$
  6. Case $k = 3$
  7. Case $k = s$
  8. Implementation Issues: $k$-factor computation
  9. Complexity Analysis
  10. A web application for visualizing $k$-factors
  11. Walkthrough
  12. Conclusions

Key Result

Theorem 1

A sequence $\langle d_1 ,\ldots , d_n \rangle$ of nonnegative integers in nonincreasing order is graphic if and only if: (a)$\Sigma_{i=1}^n d_i$ even and, (b) for each integer $k$ with, $1 \leq k \leq n$, holds.

Figures (16)

  • Figure 1: A graph on dSix with a connected 2-factor
  • Figure 2: Connected 3-factor for the degree sequence (10, 10, 10, 10, 9, 9, 9, 9, 8, 8, 8 , 8, 7, 7, 7, 7, 6, 4)
  • Figure 3: A graphic realization of 6,6,6,6,5,5,5,5 with a connected 2-factor
  • Figure 4: A graph on dSix with a disconnected 2-factor
  • Figure 5: Graph for the degree sequence (15, 15, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 2, 2)
  • ...and 11 more figures

Theorems & Definitions (12)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Conjecture 1
  • Theorem 2
  • Theorem 3
  • Example 1
  • Example 2
  • Definition 3
  • Proposition 1
  • ...and 2 more