Extending an earlier work by Kostochka for subcubic graphs, we show that a connected graph $G$ with minimum degree $2$ and maximum degree $4$ has at least $75^{n_4/5+n_3/10+1/5}$ spanning trees, where $n_i$ is the number of vertices of degree $i$ in $G$, unless $G$ is the complete graph on $5$ vertices or obtained from the complete graph on $6$ vertices by deleting the edges of a perfect matching. This, in particular, allows us to determine the value of the inferior limit of the normalised number of spanning trees (introduced by Alon) over the class of connected $4$-regular graphs to be $75^{1/5}$.
@article{10_37236_12576,
author = {Jean-S\'ebastien Sereni and Zelealem B. Yilma},
title = {The number of spanning trees in 4-regular simple graphs},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {4},
doi = {10.37236/12576},
zbl = {1556.05065},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12576/}
}
TY - JOUR
AU - Jean-Sébastien Sereni
AU - Zelealem B. Yilma
TI - The number of spanning trees in 4-regular simple graphs
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/12576/
DO - 10.37236/12576
ID - 10_37236_12576
ER -
%0 Journal Article
%A Jean-Sébastien Sereni
%A Zelealem B. Yilma
%T The number of spanning trees in 4-regular simple graphs
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/12576/
%R 10.37236/12576
%F 10_37236_12576
Jean-Sébastien Sereni; Zelealem B. Yilma. The number of spanning trees in 4-regular simple graphs. The electronic journal of combinatorics, Tome 31 (2024) no. 4. doi: 10.37236/12576
