Final answer:
The post-thyroidectomy patient's increasing neck swelling is the most critical information to communicate to the surgeon, potentially indicating a hematoma requiring immediate attention.
Step-by-step explanation:
The most important information to communicate to the surgeon is that the patient has increasing neck swelling after the thyroidectomy. This could be a sign of hematoma formation, which can be a life-threatening complication if it compresses the airway. The surgeon needs to be informed immediately as it may require urgent intervention. While report of pain, sleepiness, and a high cardiac rate are also concerns, they are more common and less acutely dangerous post-operative experiences. Sleepiness can be an expected after effect of general anesthesia, and a slightly elevated heart rate could be due to various factors including pain or normal responses to surgery and may not be as immediately critical.