Geodesic
Critical Graphs for R(Pn, Pm) and the Star-Critical Ramsey Number for Paths
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 4, pp. 689-701

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The graph Ramsey number R(G,H) is the smallest integer r such that every 2-coloring of the edges of Kr contains either a red copy of G or a blue copy of H. The star-critical Ramsey number r*(G,H) is the smallest integer k such that every 2-coloring of the edges of Kr − K1,r−1−k contains either a red copy of G or a blue copy of H. We will classify the critical graphs, 2-colorings of the complete graph on R(G,H) − 1 vertices with no red G or blue H, for the path-path Ramsey number. This classification will be used in the proof of r∗(Pn, Pm).
Keywords: Ramsey number, critical graph, star-critical Ramsey number, path 
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     author = {Hook, Jonelle},
     title = {Critical {Graphs} for {R(P\protect\textsubscript{n},} {P\protect\textsubscript{m})} and the {Star-Critical} {Ramsey} {Number} for {Paths}},
     journal = {Discussiones Mathematicae. Graph Theory},
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Hook, Jonelle. Critical Graphs for R(Pn, Pm) and the Star-Critical Ramsey Number for Paths. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 4, pp. 689-701. http://geodesic.mathdoc.fr/item/DMGT_2015_35_4_a7/