3-Coloring Triangle-Free Planar Graphs with a Precolored 9-Cycle

Further Extensions of the Grötzsch Theorem

“…On the other hand, a precoloring of a 5-face in a planar graph with one triangle cannot always be extended to a 3-coloring of the whole graph; see example in Figure 3. The result about extending a precoloring of an 8-cycle from [17] (as remarked in [12]) implies the following. Theorem 1.9 ( [17]).…”

“…On the other hand, if the face f has length k with k ≥ 6, then not every precoloring of its vertices can be extended to a 3-coloring of G. The cases when k = 6, 7, 8, 9 were completely characterized in [20], [4], [17], and [12], respectively. Moreover, precoloring faces in planar graph of girth at least 5 have also been studied (see, e.g., [12] for more details).…”

3-coloring triangle-free planar graphs with a precolored 9-cycle

“…Every order, or the pair K 1 , K 2 , is assigned a kind (t 1 t 2 ), where t i is the number of vertices from {u 3−i , v 3−i } that are in P i . Hence there are only five possible kinds; namely (00), (02), (20), (22), and (11). If K 1 is the same path as K 2 , we can pick any order that will give kind (00), (02), or (20).…”

“…Suppose that the order is u 1 , v 2 , v 1 , u 2 , which gives kind (11). By planarity, there exists a vertex x such that x is an internal vertex of both K 1 and K 2 .…”

“…Lemma 9. Let G be a triangle-free plane graph with outer face C where every 4-cycle bounds a face and let K 1 and K 2 be paths in G with endpoints u 1 , v 1 , u 2 , v 2 in C. Let K 1 and K 2 be internally disjoint with C and the order of u 1 , v 1 , u 2 , and v 2 form (11). Then there exists a common point for K Proof.…”