I,F-partitions of sparse graphs

Abstract

A star k-coloring is a proper k-coloring where the union of two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an I,F-partition. We use a combination of potential functions and discharging to prove that every graph with maximum average degree less than 5 2 has an I,F-partition, which is sharp and answers a question of Cranston and West [A guide to the discharging method, arXiv:1306.4434]. This result implies that planar graphs of girth at least 10 are star 4-colorable, improving upon previous results of Bu, Cranston, Montassier, Raspaud, and Wang [Star coloring of sparse graphs, J. Graph Theory 62 (2009), 201-219].

On Star 5-Colorings of Sparse Graphs

“…In particular, is h(2) = 46 17 ? As layed out in Table 1, a planar graph with girth at least 10 is star 4-colorable [6], which is sharp in the sense that the number of colors cannot be reduced [1]. The main result in this paper implies that a planar graph with girth at least 8 is star 5-colorable.…”

“…Determining the exact values of f (k) and h(k) is a difficult, yet interesting problem. From [6,7], we know f (1) = 1, f (2) = 3 2 , f (3) = 2, and 5 2 ≤ f (4) ≤ 18 7 . Our main result implies f (5) ≥ 8 3 .…”

“…Since a graph H with mad(H) = 5 2 where H has no FI 1 -partition was constructed in [7], we know h(1) ≤ 5 2 . Yet, Brandt et al [6] proved that a graph G with mad(G) < 5 2 has an FI 1 -partition, so the value of h(1) is determined, namely, h(1) = 5 2 . In this term, our main result is equivalent to h(2) ≥ 8 3 .…”

“…See Table 1 for a summary of lower and upper bounds on the star chromatic number of a planar graph with a given girth constraint. Various results above are also true for the maximum average degree setting [6,18,21] and the list version setting [7,8,10,18]. Researchers have also investigated star coloring for bipartite planar graphs [16] and subcubic graphs [9,10].…”

“…Sufficient conditions on girth to guarantee that a planar graph is star 4-colorable have received much attention due to its relation to the Four Color Theorem [2,3]. In particular, Bu et al [7] improved the girth constraint to 13, and Brandt et al [6] has the current best bound showing that a planar graph with girth at least 10 has a star 4-coloring.…”