Let
$$c_{1},c_{2},\ldots ,c_{k}$$
be k non-negative integers. A graph G is
$$(c_{1},c_{2},\ldots ,c_{k})$$
-colorable if the vertex set can be partitioned into k sets
$$V_{1},V_{2},\ldots ,V_{k}$$
such that for every
$$i,1\le i\le k$$
, the subgraph
$$G[V_{i}]$$
has maximum degree at most
$$c_{i}$$
. Steinberg (Ann Discret Math 55:211–248, 1993) conjectured that every planar graph without 4- and 5-cycles is 3-colorable. Xu and Wang (Sci Math 43:15–24, 20...
