How to cool a graph

TL;DR

A new graph parameter called the cooling number is introduced, inspired by the spread of influence in networks and its predecessor, the burning number, which measures the speed of a slow-moving contagion in a graph.

Abstract

We introduce a new graph parameter called the cooling number, inspired by the spread of influence in networks and its predecessor, the burning number. The cooling number measures the speed of a slow-moving contagion in a graph; the lower the cooling number, the faster the contagion spreads. We provide tight bounds on the cooling number via a graph's order and diameter. Using isoperimetric results, we derive the cooling number of Cartesian grids. The cooling number is studied in graphs generated by the Iterated Local Transitivity model for social networks. We conclude with open problems.

Figures (4)

• Figure 1: An example of cooling on the cycle of length $8$. Black labels indicate the nodes of the cooling sequence in increasing order. Blue labels indicate the round that the corresponding node was cooled.
• Figure 2: An example of cooling on the complete caterpillar of length 6. Black labels indicate the nodes of the cooling sequence in increasing order. Blue labels indicate the round that the corresponding node was cooled.
• Figure 3: An example of the cooling sequence for the $15 \times 15$ grid using the strategy of Theorem \ref{['thm:cooling_grids_main']}. The nodes on the first row with distance $d_2,d_3,d_4$ from $(1,1)$ are also indicated.
• Figure 4: An example of cooling on $\mathrm{ILT}(P_6)$. Black labels indicate the nodes of the cooling sequence in increasing order. Blue labels indicate the round that the corresponding node was cooled.

Theorems & Definitions (24)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Lemma 5: kvis
• proof