[Paper Review] (1,0,0)-colorability of planar graphs without cycles of length 4 or 6

Ligang Jin, Yingli Kang+2 more|arXiv (Cornell University)|Jan 1, 2020

Computational Geometry and Mesh Generation23 references3 citations

TL;DR

This paper proves that every planar graph without 4-cycles or 6-cycles is (1,0,0)-colorable, meaning its vertices can be partitioned into three sets where one set induces a subgraph of maximum degree at most 1, and the other two induce subgraphs with maximum degree 0. The result improves upon prior (2,0,0)- and (1,1,0)-colorability results using a discharging method with reducible configurations and charge redistribution rules, resolving a key case in the study of improper colorings of planar graphs without short cycles.

ABSTRACT

A graph $G$ is $(d_1,d_2,d_3)$-colorable if the vertex set $V(G)$ can be partitioned into three subsets $V_1,V_2$ and $V_3$ such that for $i\in\{1,2,3\}$, the induced graph $G[V_i]$ has maximum vertex-degree at most $d_i$. So, $(0,0,0)$-colorability is exactly 3-colorability. The well-known Steinberg's conjecture states that every planar graph without cycles of length 4 or 5 is 3-colorable. As this conjecture being disproved by Cohen-Addad etc. in 2017, a similar question, whether every planar graph without cycles of length 4 or $i$ is 3-colorable for a given $i\in \{6,\ldots,9\}$, is gaining more and more interest. In this paper, we consider this question for the case $i=6$ from the viewpoint of improper colorings. More precisely, we prove that every planar graph without cycles of length 4 or 6 is (1,0,0)-colorable, which improves on earlier results that they are (2,0,0)-colorable and also (1,1,0)-colorable, and on the result that planar graphs without cycles of length from 4 to 6 are (1,0,0)-colorable.

Motivation & Objective

• To resolve whether planar graphs without cycles of length 4 or 6 are (1,0,0)-colorable, advancing beyond known (2,0,0)- and (1,1,0)-colorability results.
• To extend the understanding of improper colorings in planar graphs, particularly in the context of Steinberg-type conjectures and their relaxations.
• To establish a stronger result than previous work showing (1,0,0)-colorability for planar graphs excluding cycles of length 4 to 6.
• To apply the discharging method with carefully designed charge redistribution rules to prove the main theorem via structural analysis of reducible configurations.

Proposed method

• The proof uses the discharging method, assigning initial charges to vertices and faces based on degree, then redistributing them via predefined rules to show final charges are non-negative.
• Key rules include transferring charge 1 from 8+-faces to incident 2-vertices (R11), and distributing charge from vertices to faces based on face type and incident vertex degrees.
• The proof identifies and proves reducibility of specific configurations, such as faces with internal 2-vertices or specific cycle structures, to eliminate potential counterexamples.
• Structural lemmas (e.g., Lemma 3.1, 3.7, 3.19, 3.20) are used to constrain the local structure of faces and vertices, especially regarding 2-vertices and heavy neighbors.
• The discharging rules are tailored to handle different face types (3-, 5-, and 8+-faces) and vertex degrees, with special handling for weak/strong faces and antiwheels.
• A super-extended theorem is formulated and proven, which strengthens the main result by incorporating additional structural constraints on the graph.

Experimental results

Research questions

• RQ1 Is every planar graph without cycles of length 4 or 6 (1,0,0)-colorable?
• RQ2 Can the (1,0,0)-colorability result be established for planar graphs excluding 4- and 6-cycles, improving upon prior (2,0,0)- and (1,1,0)-colorability results?
• RQ3 What structural properties of planar graphs without 4- or 6-cycles allow for a stronger improper coloring result?
• RQ4 Are there reducible configurations in such graphs that can be used to prove (1,0,0)-colorability via discharging?

Key findings

• Every planar graph without cycles of length 4 or 6 is (1,0,0)-colorable, which is a stronger result than the previously known (2,0,0)- and (1,1,0)-colorability.
• The proof establishes that 8+-faces send charge 1 to each incident 2-vertex, ensuring non-negative final charge for such faces.
• For 5-faces, the final charge is at least 0, achieved by receiving at least 8/3 from a 5+-vertex or sufficient charge from multiple 4+-vertices.
• All 3-faces have final charge exactly 0, achieved through charge reception from incident 3-vertices, 4-vertices, and heavy outer neighbors based on face type and strength.
• The discharging process confirms that no face or vertex ends with negative charge, validating the reducibility and structural assumptions.
• The result improves on prior work showing (1,0,0)-colorability for planar graphs excluding cycles of length 4 to 6, now extending to the case excluding only 4- and 6-cycles.