Note on 3-choosability of planar graphs with maximum degree 4 (original) (raw)

3-Choosability of Triangle-Free Planar Graphs with Constraints on 4-Cycles

SIAM Journal on Discrete Mathematics, 2010

A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. A theorem by Grötzsch [2] asserts that every triangle-free planar graph is 3-colorable. On the other hand Voigt [10] found such a graph which is not 3-choosable. We prove that if a triangle-free planar graph is not 3-choosable, then it contains a 4-cycle that intersects another 4-or 5-cycle in exactly one edge. This strengthens the Thomassen's result [8] that every planar graph of girth at least 5 is 3-choosable. In addition, this implies that every triangle-free planar graph without 6-and 7-cycles is 3-choosable.

A note on the acyclic 3-choosability of some planar graphs

Discrete Applied Mathematics, 2010

An acyclic coloring of a graph G is a coloring of its vertices such that : (i) no two adjacent vertices in G receive the same color and (ii) no bicolored cycles exist in G. A list assignment of G is a function L that assigns to each vertex v ∈ V (G) a list L(v) of available colors. Let G be a graph and L be a list assignment of G. The graph G is acyclically L-list colorable if there exists an acyclic coloring φ of G such that φ(v) ∈ L(v) for all v ∈ V (G). If G is acyclically L-list colorable for any list assignment L with |L(v)| ≥ k for all v ∈ V (G), then G is acyclically k-choosable. In this paper, we prove that every planar graph with neither cycles of lengths 4 to 7 (resp. to 8, to 9, to 10) nor triangles at distance less 7 (resp. 5, 3, 2) is acyclically 3-choosable.

A sufficient condition for planar graphs to be 3-colorable

Journal of Combinatorial Theory, 2003

Planar graphs without 3-cycles at distance less than 4 and without 5-cycles are proved to be 3-colorable. We conjecture that, moreover, each plane graph with neither 5-cycles nor intersecting 3-cycles is 3-colorable. In this conjecture, none of the two assumptions can be dropped because there exist planar 4-chromatic graphs without 5-cycles, as well as planar 4chromatic graphs without intersecting triangles. r

Some structural properties of planar graphs and their applications to 3-choosability

Discrete Mathematics, 2012

In this article, we consider planar graphs in which each vertex is not incident to some cycles of given lengths, but all vertices can have different restrictions. This generalizes the approach based on forbidden cycles which corresponds to the case where all vertices have the same restrictions on the incident cycles. We prove that a planar graph G is 3-choosable if it is satisfied one of the following conditions: (1) each vertex x is neither incident to cycles of lengths 4, 9, ix with ix ∈ {5, 7, 8}, nor incident to 6-cycles adjacent to a 3-cycle. (2) each vertex x is not incident to cycles of lengths 4, 7, 9, ix with ix ∈ {5, 6, 8}. This work implies five results already published [13, 3, 7, 12, 4].

Planar graphs without adjacent cycles of length at most seven are 3-colorable

Discrete Mathematics, 2010

We prove that every planar graph in which no i-cycle is adjacent to a j-cycle whenever 3 ≤ i ≤ j ≤ 7 is 3-colorable and pose some related problems on the 3-colorability of planar graphs. 1 Introduction In 1976, Appel and Haken proved that every planar graph is 4-colorable [3, 4], and as early as 1959, Grötzsch [15] proved that every planar graph without 3-cycles is 3-colorable. As proved by Garey, Johnson and Stockmeyer [14], the problem of deciding whether a planar graph is 3-colorable is NP-complete. Therefore, some sufficient conditions for planar graphs to be 3-colorable were stated. In 1976, Steinberg [19] raised the following: Steinberg's Conjecture '76 Every planar graph without 4-and 5-cycles is 3-colorable. In 1969, Havel [16] posed the following problem: Havel's Problem '69 Does there exist a constant C such that every planar graph with the minimum distance between triangles at least C is 3-colorable?

Planar graphs without triangles adjacent to cycles of length from 4 to 7 are 3-colorable

Discrete Mathematics, 2010

It is known that planar graphs without cycles of length from 4 to 7 are 3-colorable (Borodin et al., 2005) [13] and that planar graphs in which no triangles have common edges with cycles of length from 4 to 9 are 3-colorable (Borodin et al., 2006) [11]. We give a common extension of these results by proving that every planar graph in which no triangles have common edges with k-cycles, where k ∈ {4, 5, 7} (or, which is equivalent, with cycles of length 3, 5 and 7), is 3-colorable.

On the Three Colorability of Planar Graphs

In this paper we have given a new three colorability criteria for planar graphs that can be considered as an generalization of the Heawood and the Grotszch theorems with respect to the triangulation and cycles of length greater than ≥ 4. We have shown that an triangulated planar graph with k disjoint holes is 3-colorable if and only if every hole satises the parity symmetric property, where a hole is a cycle (face boundary) of length greater than 3 or an induced outerplanar subgraph.

Planar graphs without 5- and 7-cycles and without adjacent triangles are 3-colorable

Journal of Combinatorial Theory, Series B, 2009

It is known that planar graphs without cycles of length from 4 to 7 are 3-colorable [O.V. Borodin, A.N. Glebov, A. Raspaud, M.R. Salavatipour, Planar graphs without cycles of length from 4 to 7 are 3-colorable, J. Combin. Theory Ser. B 93 (2005) 303-311]. We improve this result by proving that every planar graph without 5-and 7-cycles and without adjacent triangles is 3-colorable. Also, we give counterexamples to the proof of the same result in [B. Xu, On 3-colorable plane graphs without 5-and 7-cycles, J. Combin. Theory Ser. B 96 (2006) 958-963].

(3, 1)*-CHOOSABILITY of Planar Graphs Without Adjacent Short Cycles

Electronic Notes in Discrete Mathematics, 2013

A list assignment of a graph G is a function L that assigns a list L(v) of colors to each vertex v ∈ V (G). An (L, d) *-coloring is a mapping π that assigns a color π(v) ∈ L(v) to each vertex v ∈ V (G) so that at most d neighbors of v receive color π(v). A graph G is said to be (k, d) *choosable if it admits an (L, d) *-coloring for every list assignment L with |L(v)| ≥ k for all v ∈ V (G). In 2001, Lih et al. [6] proved that planar graphs without 4and l-cycles are (3, 1) *choosable, where l ∈ {5, 6, 7}. Later, Dong and Xu [3] proved that planar graphs without 4and l-cycles are (3, 1) *-choosable, where l ∈ {8, 9}. There exist planar graphs containing 4-cycles that are not (3, 1) *-choosable (Crown, Crown and Woodall, 1986 [1]). This partly explains the fact that in all above known sufficient conditions for the (3, 1) *-choosability of planar graphs the 4-cycles are completely forbidden. In this paper we allow 4-cycles nonadjacent to relatively short cycles. More precisely, we prove that every planar graph without 4-cycles adjacent to 3and 4-cycles is (3, 1) *-choosable. This is a common strengthening of all above mentioned results. Moreover as a consequence we give a partial answer to a question of Xu and Zhang [11] and show that every planar graph without 4-cycles is (3, 1) *choosable.