Currently there is a record for the second Fulton v Riley bout dated 2/2/1914. In fact, that fight took place on about 4/18/1913. I haven't found a date for that first match yet, but it was certainly early in 1913 or late in 1912.
Per the Minneapolis Tribune on 20 April 1913 (Sport Section, Page 4), it notes that the first time the two met, Fulton handed Riley easily for four rounds before Riley battled back to win the last two rounds - but not enough to take a decision from Fulton or even hurt him.
In the second bout in April, Fulton won by knockout in the second round in a fight scheduled for ten rounds.
"From the jump it was evident that Riley was to be outclassed, for Fulton outboxed him and hit him as he pleased. Riley gamely attempted to rush things in the opening session but Fulton kept him away with hard left hand jabs and right crosses. In the second round, too, Fulton punished his man severely and managed to keep Riley away, although the latter tried continually to get into close quarters. The round hadn't gone very far before Fulton landed a terrific straight right to the body, which he followed with straight lefts to the face and these did the business."