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Exact distance Kneser graphs

Agustina Victoria Ledezma, Adrián Pastine, Mario Valencia-Pabon

Abstract

For any graph $G = (V,E)$ and positive integer $d$, the exact distance-$d$ graph $G_{=d}$ is the graph with vertex set $V$, where two vertices are adjacent if and only if the distance between them in $G$ is $d$. We study the exact distance-$d$ Kneser graphs. For these graphs, we characterize the adjacency of vertices in terms of the cardinality of the intersection between them. We present formulas describing the distance between any pair of vertices and we compute the diameter of these graphs.

Exact distance Kneser graphs

Abstract

For any graph and positive integer , the exact distance- graph is the graph with vertex set , where two vertices are adjacent if and only if the distance between them in is . We study the exact distance- Kneser graphs. For these graphs, we characterize the adjacency of vertices in terms of the cardinality of the intersection between them. We present formulas describing the distance between any pair of vertices and we compute the diameter of these graphs.
Paper Structure (7 sections, 29 theorems, 59 equations)

This paper contains 7 sections, 29 theorems, 59 equations.

Table of Contents

  1. Introduction and main results
  2. Adjacency characterization
  3. Computing the distance function
  4. Case $d=2p$
  5. Case $d = 2p+1$
  6. Computing the diameter
  7. Acknowledgments

Key Result

Lemma 1.1

Let $A,B \in [2k+r]^k$ be two vertices in $K(2k+r,k)$ joined by a path of length $2p$ ($p \geq 0$). Then $|A \cap B| \geq k-rp$.

Theorems & Definitions (49)

  • Lemma 1.1: STA76
  • Corollary 1.2
  • Proposition 1.3: VV05
  • Lemma 1.4: VV05
  • Theorem 1.5: VV05
  • Theorem 1.6: CheW08
  • Theorem 1.7: CheW08
  • Theorem 1.8: AAC18
  • Theorem 1.9: AAC18
  • Theorem 1.10
  • ...and 39 more