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Partite saturation number of cycles

Yiduo Xu, Zhen He, Mei Lu

Abstract

A graph $H$ is said to be $F$-saturated relative to $G$, if $H$ does not contain any copy of $F$, but the addition of any edge $e$ in $E(G)\backslash E(H)$ would create a copy of $F$. The minimum size of an $F$-saturated graph relative to $G$ is denoted by $sat(G,F)$. Let $K_k^n$ be the complete $k$-partite graph containing $n$ vertices in each part and $C_\ell$ be the cycle of length $\ell$. In this paper we give an asymptotically tight bound of $sat(K_k^n,C_\ell)$ for all $ \ell \geq 4, k \geq 2$ except $(\ell,k)=(4,4)$. Moreover, we determined the exact value of $sat(K_k^n,C_\ell)$ for $ k>\ell=4 $ and $5 \geq \ell>k \geq 3$ and $(\ell,k)=(6,2)$.

Partite saturation number of cycles

Abstract

A graph is said to be -saturated relative to , if does not contain any copy of , but the addition of any edge in would create a copy of . The minimum size of an -saturated graph relative to is denoted by . Let be the complete -partite graph containing vertices in each part and be the cycle of length . In this paper we give an asymptotically tight bound of for all except . Moreover, we determined the exact value of for and and .

Paper Structure

This paper contains 11 sections, 38 theorems, 22 equations, 21 figures, 1 table.

Table of Contents

  1. Introduction
  2. Bipartite saturation number of even cycles
  3. Constructions of saturated subgraphs of $K_{k}^{ n}$
  4. Construction I
  5. Construction II
  6. Construction III
  7. $C_4$-saturation multipartite graph
  8. Tripartite
  9. Non-tripartite
  10. $C_5$-saturation multipartite graph
  11. Concluding remark

Key Result

Theorem 1.1

For $\ell \geq 3$ and $n_1,n_2 \geq \ell+2$, $sat(K_{n_1,n_2},C_{2 \ell}) \leq n_1+n_2 + \ell^2 - 3 \ell +1$. Moreover, $sat(K_{n_1,n_2},C_{6}) = n_1+n_2 + 1$.

Figures (21)

  • Figure 1: $\mathscr{G}_{n_1,n_2}^{\ell} = V_1 \cup V_2$. $V_i = B_i \cup A_i$, $B_2'=B_2 \setminus \{y_1\}$ such that $|B_1|=\ell$, $|B_2'|=\ell-1$. The solid line represents the complete connection between vertices.
  • Figure 2: $W^{(4,3,n)} = V_1 \cup V_2 \cup V_3$, $V_i = B_i \cup \{x_i \}$, $A=\{x_1,x_2,x_3\}$. The solid line represents the complete connection between vertices, and the dotted ellipse represents $W^{(4,3,n)}[A]= \gamma^{(4,3)}=K_3$.
  • Figure 3: $W_{\pi_1}^{(5,4,n)}$ (left) , $W_{\pi_2}^{(5,4,n)}$ (right). $W_{\pi_i}^{(5,4,n)} = V_1 \cup V_2 \cup V_3 \cup V_4$, $V_i = B_i \cup \{x_i \}$, $A=\{x_1,x_2,x_3,x_4\}$. The solid line represents the complete connection between vertices, and the dotted ellipse represents $W_{\pi_i}^{(5,4,n)}[A]= \gamma^{(5,4)}=K_4$.
  • Figure 4: $W_{\pi}^{(6,5,n)} = V_1 \cup V_2 \cup V_3 \cup V_4 \cup V_5$, $V_1 = B_1 \cup \{x_1,x_2 \}$, $V_2 = B_2 \cup \{x_3,x_4 \}$, $V_3 = B_3 \cup \{x_5 \}$, $V_4=B_4$, $V_5=B_5$, $A=\{x_1,\ldots,x_5\}$. The solid line represents the complete connection between vertices, and the dotted pentagon represents $W_{\pi}^{(6,5,n)}[A]= \gamma^{(6,5)}=K_{2,2,1}$.
  • Figure 5: $W_{*}^{(5,3,n)} = V_1 \cup V_2 \cup V_3$, $|V_i|=n$. $V_1 = B_1 \cup \{u^*,w \}$, $V_2 = B_2 \cup \{u_1,u_3\}$, $V_3 = B_3 \cup \{u_2 \}$. The solid line represents the complete connection between vertices.
  • ...and 16 more figures

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 38 more