Three-coloring triangle-free planar graphs in linear time (SODA 09’) Zdenek Dvorak, Ken-ichi Kawarabayashi, Robin Thomasz Presented By Chen Yang

Basic Concept Planar Graph Triangle-free graph no subgraph that is isomorphic to or is a subdivision of K5 or K3, 3 Triangle-free graph no three vertices forming a triangle of edges A finite graph is planar if and only if it contains no subgraph that is isomorphic to or is a subdivision of K5 or K3, 3 where K5 is the complete graph of order 5 and K3­ is the complete bipartite graph with 3 vertices in each of the halves. A triangle-free graph is an undirected graph. In the graph, there are no three vertices forming a triangle of edges. Or formally, Triangle-free graphs can be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs.

Triangle-Free Planar Graph Sample

Grötzsch's theorem Every triangle-free planar graph can be colored with only 3 colors. Not every non-planar triangle-free graph is 3-colorable. Cannot be generalized to all planar K4-free graphs.

Related Work Thomassen presented an quadratic time algorithm. J. Gimbel and C. Thomassen, Coloring graphs with fixed genus and girth, Trans. Amer. Math. Soc., 349 (1997), pp. 4555--4564. Kowalik and Kurowski achieved an O (nlogn) time by providing a sophisticated structure. Lukasz Kowalik: Fast 3-Coloring Triangle-Free Planar Graphs. ESA 2004: 436-447 Lukasz Kowalik, Maciej Kurowski: Oracles for bounded- length shortest paths in planar graphs. ACM Transactions on Algorithms 2(3): 335-363 (2006)

Introduction This paper presents a linear time algorithm for three coloring triangle-free planar graph. It avoids any complex data structure and easy to implement.

How Exhibit five reducible configurations, called “multigram” Show that every triangle-free planar graph G contains one of those reducible configurations. Look for a reducible configuration in G Modify G to a smaller graph G’, that is r-reduction State every 3-coloring of G’ can be converted into a 3-coloring of G in constant time. Apply algorithm recursively to G’ to get a 3-coloring

Reducible Configurations Tetragram A sequence (v1, v2, v3, v4) of vertices which form a facial cycle in G in the order listed. Hexagram A sequence, (v1, v2, v3…, v6) is defined similarly. Pentagram A sequence (v1, v2, v3, v4, v5) of vertices which form a facial cycle in G in the order listed v1, v2, v3, v4 all have degree exactly 3

Reducible Configurations Decagram: A pentagram (v1,v2..v5) is called a decagram if v5 has degree exactly three. Monogram: A sequence (v) comprised of a vertex v belonging to V(G) of degree at most two.

Lemma 2.1 Let G be a connected triangle-free plane graph and let f0 be the unbounded face f0 of G. Assume that the boundary of f0 is a cycle C of length at most six, and that every vertex of G not on C has degree at least three. If G doesn’t equal C, then G has either a tetragram or a pentagram (v1, v2, v3, v4, v5) such that v1, v2, v3, v4 are not in V(C).

Safe Tetragram & Hexagram Tetragram: it is safe if every path in G of length at most three with ends v1 and v3 if a subgraph of the cycle v1v2...vk. Hexagram: the safety definition is the same with Tetragram.

Safe Pentagram For i=1,2,3,4, let xi be the neighbor of vi distinct from vi-1 and vi+1(where v0 = v5 ). Then xi that are not in {v1…v5} because G is triangle-free. The vertices x1, x2, x3, v4 are pairwise distinct and pairwise non-adjacent. No path in G \ {v1, v2, v3, v4} of length at most three from x2 to x5. Every path in G\{v1,v2,v3,v4} of length at most three from x3 to x4 has length exactly two, and its completion via the path x3v3v4x4 results in a facial cycle of length five in G. in particular, there is at most one such path.

Lemma 2.2 Every triangle-free plane graph G of minimum degree at least three has a safe tetragram, pentagram or hexagram.

By-Product: Proof of Grötzsch's Theorem Lemma 2.1 makes sure that the connected triangle-free plane graph has either a tetragram, or a pentagram. Define what is safe tetra, penta and hexagram. Lemma 2.2 states that every triangle-free plane graph G of minimum degree at least 3 has a safe tetra-, penta- or hexagram. Proof if G contains safe tetra-, penta- or hexagram, it will be 3- colorable. If k = 4 or k = 6, then apply induction to the graph obtained from G by identifying v1 and v3. It follows from the definition of safety that the new graph has no triangles, and clearly every 3-coloring of the new graph extends to a 3-coloring of G.

Constant Time Operations(D=47 here) Remove an edge. Add an edge, assuming that the edges preceding and following it in the facial walks are specified. Remove an isolated vertex. Determine the degree of a vertex v if deg(v) ≤ D or prove that deg(v) > D Check whether two vertices u and v such that min (deg(u), dev(v)) ≤ D are adjacent. Check whether the distance between two vertices u and v such that max(deg(u), deg(v)) ≤ D is at most two. Given an edge e of a face f, either output all vertices incident with the component of the boundary of f that contains e, or establish that the length of f is strictly larger than six. Output the subgraph consisting of vertices which is reachable from a vertex v0 through a path v0, v1…. vt of length t ≤ D, such that deg (vi) ≤ D for 0≤i≤t.

Essential Concepts Two essential concepts were introduced, C-admissible and C-forbidden under these assumptions: G: plane graph k ∈ {1, 4, 5, 6} r = (v1, v2... vk) be a mono, tera, penta or hexgram in G. C is a subgraph of G. A vertex of G is big if it has degree at least 48, and small otherwise. A vertex v belonging to V(G) is C-admissible if it is small and does not belong to C; otherwise it is C-forbidden.

Lemma 4.1 Suppose G is a triangle-free plane graph, r is a safe multigram in G, and let G’ be the r-reduction of G. Then G’ is triangle-free, and every 3-coloring of G’ can be converted to a 3-coloring of G in constant time.

r-reduction If r is a monogram, it is always safe G’:=G\v1 it is C-secure if v1 doesn’t belong V(C). G’:=G\v1

r-reduction If r is a tetragram, it is safe if the only paths in G of length at most three with ends v1 and v3 are subgraphs of facial cycle v1v2v3v4. if r is safe, v1 is C-admissible and has degree exactly three, r is C-secure. Letting x denote the neighbor of v1 other than v2 and v4, the vertex x is C-admissible and either v3 is C-admissible, or every neighbor of x is C-admissible. G’ is obtained from G by identifying the vertices v1 and v3 and deleting one edge from each of the two pairs of parallel edges.

r-reduction If r is a decagram For i = 1, 2, 3, 4 let xi be the neighbor of vi other than vi-1 or vi+1, v0 means v5. r is safe if x1, x3 are distinct, non-adjacent and there is no path of length two between them. Also r is C-secure if it is safe and the vertices v1, v2 … v5, x1, x3 are all C- admissible. G’ is obtained from G\{v1,v2…v5} by adding the edge x1x3.

r-reduction If r is a pentagram, r is C-secure if it is safe, the vertices v1, v2 … v5, x1, x2, x3, x4 are all C-admissible, either v5 or x2 has not C-forbidden neighbor and either x3 or x4 has no C-forbidden neighbor. G’ is obtained from G\{v1,v2,v3,v4} by identifying x2 and v5; identifying x3 and x4; and deleting one of the parallel edges should x3 and x4 have a common neighbor.

r-reduction If r is a hexagram r is safe if every path of length at most three in G between v1 and v3 is the path v1v2v3. r is C-secure if v1, v3, v6 are C-admissible, v1 has degree exactly three, and the neighbor of v1 other than v2 or v6 is C-admissible. G’ is obtained from G be identifying the vertices v1 and v3 and deleting one of the parallel edges

Algorithm 1 Algorithm 4.1. There is an algorithm with following specifications: Input: A triangle-free planar graph. Output: A proper 3-coloring of G. Running time: O(|V(G)|).

Description for Algorithm 1 Assume that G is a plane graph. The algorithm is recursive. maintain a list L that will include the pivots of all secure multigrams in G. Initializing the list L to consist of all vertices of G of degree at most three. Iteration starts. Remove a vertex from L. If v∉V(G), go to the next iteration Otherwise, check if G has a secure multigram with pivot v. (this can be performed in constant time) If no such multigram exist, go to the next iteration. Otherwise, let r e one such multigram, and let G’ be the r-reduction of G. G’ is triangle-free (lemma 4.1) and can be constructed in constant time by adding and deleting bounded number of edges. Apply the algorithm recursively to G’ and converts the resulting 3-coloring of G’ to one of G.

Lemma 4.2 Given a triangle-free plane graph G and a vertex v∈V(G), it can be decided in constant time whether G has a secure multigram with pivot v. The vertex v1 will be called the pivot of the multigram (v1; v2; ….; vk).

Algorithm 2 Input: A triangle-free plane graph G, a facial cycle C in G of length at most five and a proper 3-coloring ∅ of C. Output; A proper 3-coloring of G whose restriction to V(C) is equal to ∅. Running time: O(|V(G)|).

Description for Algorithm 2 Same as Algorithm 1, but use “C-secure” rather than “secure”

Lemma 5.1 This lemma supports the correctness of Algo 2. Let G be a connected triangle-free plane graph and let f0 be its outer face. Assume that f0 is bounded by a cycle of length at most six and that |V(G) – V(C)| ≥ 2. Then either G has a C-secure multigram, or C has length exactly six and includes at least two distinct non-adjacent C-appendices.

Further works of authors since this paper Zdenek Dvorak and Robin Thomas Three-coloring triangle-free graphs on surfaces I. Extending a coloring to a disk with one triangle CoRR abs/1010.2472 It proved that a 3-coloring of C does not extend to a 3- coloring of G if and only if C has length exactly six.