Some star-critical Ramsey numbers ?
Let Sk=K1,k S k = K 1 , k be a star of k k edges, and let Kn?Sk K n ? S k be a graph obtained from Kn K n and an additional vertex v v by joining v v and k k vertices of Kn K n . For graphs G G and H H , let r=r(G,H) r = r ( G , H ) be the Ramsey number of G G and H H . The star-critical Ramsey number r?(G,H) r ? ( G , H ) is the smallest k k such that every red/blue edge-coloring of Kr?1?Sk K r ? 1 ? S k contains a red G G or a blue H H . In this note, we show that r?(Kn,mK2)=n+2m?3 r ? ( K n , m K 2 ) = n + 2 m ? 3 for m?1 m ? 1 and n>2 n > 2 , r?(Fn,K3)=2n+2 r ? ( F n , K 3 ) = 2 n + 2 for n?2 n ? 2 , where Fn F n is an n n -fan. We also show that r?(nK4,mK3)=4n+2m r ? ( n K 4 , m K 3 ) = 4 n + 2 m for n?m?1 n ? m ? 1 and n?2 n ? 2 , and r?(nK4,mK3)=3n+3m r ? ( n K 4 , m K 3 ) = 3 n + 3 m for m?n?2 m ? n ? 2 .
star-critical ramsey number graph ramsey number critical graph