 Solution

A triangle $x_1{y}_1{y}_2$ forces at least three vertex colours. Three suffice: colour $x_1,x_2$ blue; $y_1,y_3,y_4$ orange; and $y_2,y_5$ green, as in the drawing. Hence the chromatic number is three.

The five edges incident to $x_1$ force at least five edge colours. An explicit proper five-edge-colouring is

Give $y_1{y}_2$ colour 5 and $y_1{y}_5$ colour 3. At $y_1$ the incident colours are $1,2,5,3$; at $y_2$ they are $2,3,5$; at $y_5$ they are $5,1,3$; all are distinct. The two remaining vertices each have two distinct incident colours. Thus the edge chromatic number is five.