Titus to 3 John
Provisioning the sent.
Provisioning
Paul tells Titus to make sure Zenas and Apollos have what they need for their journey as they leave Crete. Similarly, Gaius, in Third John, is praised by the elder for the way he received, and then sent with provision, some traveling brothers.
The tenses of these passages work well together. In Titus Paul anticipates an eventual departure of Zenas and Apollos, so he reminds Titus to make sure they have what they need before they go. In Third John the brothers have already left Gaius, who had given them provision, so the elder praises him for his actions.
The Book Chain may identify who the brothers were that visited Gaius. Can you think of a pair of brothers who travel around for Jesus and tend to leave a place with provision? Zenas and Apollos come to mind. They left Crete for somewhere with provision. Maybe they went to Gaius. Not hard to imagine if difficult to prove.
Titus mentions that people should be taught to do good works in times of emergency. This statement seems odd given the way it is saddled with the exhortation to provision Zenas and Apollos. What was the emergency? Paul does not specify anything going wrong. In Third John the good work that happened was the way Gaius provisioned the brothers. The emergency in Third John is the way Diotrephes did not receive the brothers and even kicked out of the church people who were in favor of receiving the brothers. Likely, the provisions the brothers needed were tied up in the church, so the good work in time of emergency was Gaius' ability to personally cover their needs.
Slander
Titus and Third John share a theme of slander.
The passages do not use the word slander, but the idea of talking someone down in ways that are not factual is clearly in sight. Paul warns Titus to be careful about his choice of words in teaching so as not to give someone intending harm an easy soundbite that could be twisted. In Third John Diotrephes was "gossiping maliciously" about the elder and others. That means he was saying things that were not true.
Shun
Titus is told to shun the heretic after attempting to correct him a time or two. In Third John Diotrephes was wrongly shunning people.
Diotrephes did not receive the brothers that Gaius hosted and sent on their way with provision. And, to make matters worse, he cast out of the church those who wanted to receive the brothers. So Diotrephes was shunning many people. It's an opposit of sorts to what Titus was told, since Diotrephes was shunning people who were not heretics. The elder is clear that Gaius was in the truth and right to receive the brothers, so Diotrephes was wrong to reject them and to cast out those who wanted to receive them. Thus, Diotrephes was working against the truth, and might be the heretic that was to be shunned if he did not change after several attempts at correction.