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On Generalized Token Graphs

Xiaodi Song, Cristina Dalfó, Miquel Àngel Fiol, Mercè Mora, Shenggui Zhang

Abstract

The vertices of a $k$-token graph of a graph $G$ correspond to $k$ indistinguishable tokens placed on $k$ different vertices of $G$. Changing some conditions on both the nature of the tokens and the number of tokens allowed in each vertex of $G$, we define a generalization of token graphs, which we call generalized token graphs or simply supertoken graphs, which have different applications. Depending on the above conditions, different families of graphs (such as the Cartesian $k$-th power of $G$ by itself) are obtained, and we present some of their properties, including order, size, and connectivity.

On Generalized Token Graphs

Abstract

The vertices of a -token graph of a graph correspond to indistinguishable tokens placed on different vertices of . Changing some conditions on both the nature of the tokens and the number of tokens allowed in each vertex of , we define a generalization of token graphs, which we call generalized token graphs or simply supertoken graphs, which have different applications. Depending on the above conditions, different families of graphs (such as the Cartesian -th power of by itself) are obtained, and we present some of their properties, including order, size, and connectivity.

Paper Structure

This paper contains 10 sections, 10 theorems, 61 equations, 6 figures, 7 tables.

Table of Contents

  1. Definition and particular cases
  2. Generalized token graphs
  3. The token graphs $F_k^1(G)=F_k(G)$
  4. Isomorphism between the token and supertoken graphs of paths
  5. The supertoken graphs $F_k^s(G)$
  6. The supertoken graphs $F_k^k(G)$
  7. The supertoken graphs $F^{1}_{k\times 1}(G)$
  8. The supertoken graphs of paths and cycles
  9. The supertoken graphs $F^{s}_{k\times 1}(G)$
  10. The Cartesian product $F_{k\times 1}^k(G)\cong G\Box\stackrel{(k)}{\cdots}\Box G$

Key Result

Proposition 2.1

Let $F_{k}^s(G)$ and $F^{s}_{k\times 1}(G)$ be two $k$-supertoken graphs of a connected graph $G$ with order $n$, where $1\le s\le k$. Then, the following statements hold. $\blacktriangleleft$$\blacktriangleleft$

Figures (6)

  • Figure 1: $(a)$ The graph $C_4$; $(b)$ The supertoken graph $F_2^1(C_4)=F_2(C_4)\cong K_{2,4}\subset F_2^2(C_4)$; $(c)$ The supertoken graph $F_2^2(C_4)$; $(d)$ The supertoken graph $F_{2\times 1}^1(C_4)\subset F_{2\times 1}^2(C_4)$; $(e)$ The supertoken graph $F_{2\times 1}^2(C_4)\cong C_4\Box C_4$. The tokens are white or gray, and the rhombuses in $(b)$--$(e)$ represent the vertices of the supertoken graphs.
  • Figure 2: A connected graph $G$ with maximum degree $\Delta(G)=5$.
  • Figure 3: The graphs $F_2^1(P_7)$ (left) and $F_5^5(P_3)$ (right).
  • Figure 4: The graph $F_3^2(C_4)$.
  • Figure 5: The supertoken graph $F^{1}_{3\times 1}(P_4)$.
  • ...and 1 more figures

Theorems & Definitions (24)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Example 2.3
  • Proposition 2.4
  • proof
  • Lemma 3.1
  • proof
  • Proposition 4.1
  • ...and 14 more