On Extremal Nonsupereulerian Graphs with Clique Number \(m\)

A graph \(G\) is supereulerian if it contains a spanning eulerian subgraph. Let \(n\), \(m\), and \(p\) be natural numbers, \(m, p \geq 2\). Let \(G\) be a \(2\)-edge-connected simple graph on \(n > p + 6\) vertices containing no \(K_{m+1}\). We prove that if

\[|E(G)\leq \binom{n-p+1-k}{2}+(m-1)\binom{k+1}{2}+2p-4, \quad (1)]\

where \(k = \lfloor\frac{n-p+1}{m}\rfloor\), then either \(G\) is supereulerian, or \(G\) can be contracted to a non-supereulerian graph of order less than \(p\), or equality holds in (1) and \(G\) can be contracted to \(K_{2,p-2}\) (p is odd) by contracting a complete \(m\)-partite graph \(T_{m,n-p+1}\) of order \(n – p + 1\) in \(G\). This is a generalization of the previous results in [3] and [5].