A Ramsey goodness result for graphs with many pendant edges (Q2713630)

The chromatic surplus of a graph \(G\), \(s(G)\), is defined as the smallest number of vertices in any color class in any proper \(\chi (G)\)-coloring of \(G\). The upper chromatic surplus of a graph \(G\), \(\overline {s}(G)\), is similarly defined as the largest number of vertices in any color class in any proper \(\chi (G)\)-coloring of \(G\). It is shown that if \(G\) is a graph with no isolated vertices, \(H\) is a connected graph, \(V\) is a set of \(\overline {s}(G)\) vertices of \(H\), and \(j\) is sufficiently large, then \(r(G,H_j)=(\chi (G)-1)(n+j-1)+s(G)\) for some graph \(H_j\) obtained from \(H\) by adding \(j\) pendant edges joining new vertices to \(V\) (an edge is pendant if one of its vertices has degree \(1\)).