Distance edge-colourings and matchings

AbstractWe consider a distance generalisation of the strong chromatic index and the maximum induced matching number. We study graphs of bounded maximum degree and Erdős–Rényi random graphs. We work in three settings. The first is that of a distance generalisation of an Erdős–Nešetřil problem. The second is that of an upper bound on the size of a largest distance matching in a random graph. The third is that of an upper bound on the distance chromatic index for sparse random graphs. One of our results gives a counterexample to a conjecture of Skupień

Goldberg's Conjecture is true for random multigraphs

In the 70s, Goldberg, and independently Seymour, conjectured that for any multigraph $G$, the chromatic index $\chi'(G)$ satisfies $\chi'(G)\leq \max \{\Delta(G)+1, \lceil\rho(G)\rceil\}$, where $\rho(G)=\max \{\frac {e(G[S])}{\lfloor |S|/2\rfloor} \mid S\subseteq V \}$. We show that their conjecture (in a stronger form) is true for random multigraphs. Let $M(n,m)$ be the probability space consisting of all loopless multigraphs with $n$ vertices and $m$ edges, in which $m$ pairs from $[n]$ are chosen independently at random with repetitions. Our result states that, for a given $m:=m(n)$, $M\sim M(n,m)$ typically satisfies $\chi'(G)=\max\{\Delta(G),\lceil\rho(G)\rceil\}$. In particular, we show that if $n$ is even and $m:=m(n)$, then $\chi'(M)=\Delta(M)$ for a typical $M\sim M(n,m)$. Furthermore, for a fixed $\varepsilon>0$, if $n$ is odd, then a typical $M\sim M(n,m)$ has $\chi'(M)=\Delta(M)$ for $m\leq (1-\varepsilon)n^3\log n$, and $\chi'(M)=\lceil\rho(M)\rceil$ for $m\geq (1+\varepsilon)n^3\log n$.Comment: 26 page