List-Coloring Squares of Sparse Subcubic Graphs

Injective colorings of planar graphs with few colors

“…The Conjecture 2 was verified for several special classes of planar graphs, but it remains open for all values of ∆ ≥ 3, see [6]. Dvořák et al [4,5] have proved that the chromatic number of the square of a planar graph G with sufficiently large maximal degree is ∆ + 1 if the girth of G is at least seven and it is bounded by ∆+2 if the girth of G is six.…”

“…Initial charge. We assign charge to vertices and faces of G. For every v ∈ V(G), we assign an initial vertex charge ch 0 (v) = 9 5 d(v) − 6, and for every face f ∈ F(G), we assign an initial face charge ch 0 (f ) = 6 5 r(f ) − 6. Using Euler's formula, in a similar way as in the introduction, one can easily show that the total amount of charge is −12.…”

List‐coloring the square of a subcubic graph

“…Hence, we can assume that distance(u 1 , u 2 ) < 3. 3 , v 4 and the adjacent vertices not on the 4-cycle: u 1 , u 2 , u 3 , u 4 , respectively. In Case 2 of Lemma 7, we also assume that vertices u 1 and u 3 are adjacent and that vertices u 2 and u 4 are adjacent.…”

“…Use color c 1 on u i . Since |L(u i−1 )| = 2 and |L(v i+1 )\{c 1 }| ≥ 5 and |L(v i−1 )\{c 1 }| ≥ 5, we can chose color c 2 for u i−1 and color c 3 for v i+1 such that |L(v i−1 )\{c 1 , c 2 , c 3 }| ≥ 4. If c 2 = c 3 , then we use c 2 on vertices u i−1 and v i+1 ; Now excess(v i−1 ) ≥ 1 and excess(v i ) ≥ 2.…”

“…The best known upper bounds are due to Molloy and Salavatipour [15], who showed that χ(G 2 ) ≤ ⌈ 5 3 ∆⌉ + 78 (the constant 78 can be reduced for large ∆). Very recently, Havet, Heuvel, McDiarmid, and Reed [9] proved the upper bound 3 2 ∆(1 + o(1)). Better results can be obtained for special classes of planar graphs.…”