Graph colouring remains a central topic in graph theory, providing the mathematical framework for assigning colours to the elements of a graph under specific constraints. In particular, the ...

As early as 1913, planar graphs came up in a brainteaser called the three-utilities problem, published in The Strand Magazine. It asked readers to connect three houses to three utilities — water, gas ...

RICHARD W. KENYON, DAVID B. WILSON, SPANNING TREES OF GRAPHS ON SURFACES AND THE INTENSITY OF LOOP-ERASED RANDOM WALK ON PLANAR GRAPHS, Journal of the American Mathematical Society, Vol. 28, No. 4 ...

The weighted maximal planar graph (WMPG) is practically important in the laying out of facilities in modern manufacturing environments. Given a weighted complete graph, the WMPG seeks to find a ...

Research demonstrating that planar graphs without 4-cycles adjacent to triangles are DP-4-colourable has provided a valuable extension of previous list colouring results, thereby establishing ...