# A playful note on spanning and surplus edges

## TL;DR

This paper extends a breadth-first-walk construction for random graphs to include surplus edges, discusses two graph representations of the multiplicative coalescent, and introduces a canonical multi-graph, aiding understanding of scaling limits with surplus edges.

## Contribution

It introduces an extended framework for surplus edges in random graphs and compares different graph representations of the multiplicative coalescent.

## Key findings

- Extended breadth-first-walk construction includes surplus edges.
- Discussion of two graph representations with their advantages and drawbacks.
- Emergence of a canonical multi-graph for the multiplicative coalescent.

## Abstract

Consider a (not necessarily near-critical) random graph running in continuous time. A recent breadth-first-walk construction is extended in order to account for the surplus edge data in addition to the spanning edge data. Two different graph representations of the multiplicative coalescent, with different advantages and drawbacks, are discussed in detail. A canonical multi-graph of Bhamidi, Budhiraja and Wang (2014) naturally emerges. The presented framework should facilitate understanding of scaling limits with surplus edges for near-critical random graphs in the domain of attraction of general (not necessarily standard) eternal multiplicative coalescent.