A Characterization of the Graphs with High Degree Sum that are not Covered by Three Cycles

Abstract

Let \(G\) be a graph of order \(n\). In [A. Saito, Degree sums and graphs that are not covered by two cycles, J. Graph Theory 32 (1999), 51–61.], Saito characterized the graphs with \(\sigma_3(G) \geq n-1\) that are not covered by two cycles. In this paper, we characterize the graphs with \(\sigma_4(G) \geq n-1\) that are not covered by three cycles. Moreover, to prove our main theorem, we show several new results which are useful in the study of this area.