Search Results

If there is only one edge colouring with $k$ colours of a graph with chromatic index $k$, the graph is said to be "uniquely edge colourable". If you search on that phrase (with and without the second …

Take a maximum matching $M$ and a vertex $v$ not in $M$. If $v$ has a neighbour $w$ not in $M$, then $M+vw$ is a larger matching. So $v$ has a neighbour $x$ which is in an edge $xy$ of $M$. Now $M-x …

$$\pmatrix{ 2&3&0&0\\0&2&3&0\\0&0&2&3\\3&0&0&2}$$ Every swap of two columns or swap of two rows decreases the trace. However, there is a permutation putting all the 3s on the diagonal.

Take a circular ladder of odd length. Also known as the cartesian product of $C_n$ ($n$ odd) and $K_2$. For every perfect matching there are many ways to change it into a different perfect matching by …

The moments (power symmetric functions, sums of powers of the roots of) the characteristic polynomial enumerate all closed walks in the graph. Chris Godsil proved that the moments of the matching pol …

The moments of the adjacency matrix eigenvalues count closed walks in the graph, while the moments of the matching polynomial roots count tree-like closed walks. When the graph has few short cycles, a …

Yes, it is still just as hard. Given an arbitrary bipartite graph, in polynomial time you can remove every edge that is not in a perfect matching (test one edge at a time), thus reducing the problem …