This　paper　focuses　on　the　consensus　control　and　fault　estimation　problems　for　a　class　of　time-delayed　multi-agent　systems　with　Markov　switching　topologies.　Two　different　event-triggered　mechanisms　are　adopted　with　hope　to　reduce　burden　of　shared　network　and　improve　energy　efficiency.　Under　Markov　process,　by　establishing　the　consensus　control　protocol　and　designing　a　novel　adaptive　fault　estimation　observer,　the　consensus　control　and　fault　estimation　problems　are　transformed　into　two　stochastic　stability　problems　in　different　forms.　Then,　according　to　the　switching　Lyapunov　function　method　and　free-weighting　matrix　technique,　two　delay-dependent　stability　criteria　on　the　consensus　control　and　fault　estimation　are　derived,　respectively.　However,　the　two　criteria　containing　nonlinear　coupling　terms　are　not　standard　linear　matrix　inequalities　(LMIs)　and　cannot　be　solved　directly　with　the　LMI　toolbox.　In　order　to　eliminate　the　coupling　terms,　two　improved　path-following　algorithms　are　presented.　These　algorithms　depend　on　the　initial　conditions,　so　it　is　very　crucial　to　choose　the　appropriate　preset　parameters.　The　computational　complexity　is　increasing　with　the　number　of　iterations,　system　size　and　matrix　dimension,　which　is　a　fully　new　challenge　for　the　study　of　consensus　control　and　fault　estimation　of　multi-agent　systems.　Based　on　the　algorithms,　the　switching　consensus　controller　gains　and　model　gain　matrices　of　fault　estimation　can　be　efficiently　solved　out.　Finally,　a　simulation　example　of　tailless　fighter　airplanes　is　given　to　illustrate　the　practicality　and　validity　of　the　theoretical　results.　(c)　2021　Elsevier　B.V.　All　rights　reserved.