Coloring graphs without subdivisions of K 5 Xingxing Yu Orlando, Florida May 18, 2019 Xingxing Yu Coloring
The Four Color Theorem ◮ The Four Color Theorem (Appel and Haken 1977): If G is a planar graph then χ ( G ) ≤ 4. Xingxing Yu Coloring
The Four Color Theorem ◮ The Four Color Theorem (Appel and Haken 1977): If G is a planar graph then χ ( G ) ≤ 4. ◮ In 1997, Roberston, Sanders, Seymour, and Thomas gave a simpler and cleaner proof. Xingxing Yu Coloring
The Four Color Theorem ◮ The Four Color Theorem (Appel and Haken 1977): If G is a planar graph then χ ( G ) ≤ 4. ◮ In 1997, Roberston, Sanders, Seymour, and Thomas gave a simpler and cleaner proof. ◮ Theorem (Ringel and Youngs 1968): If G is embeddable in a surface other than the sphere with Euler characteristic χ ∗ then � 7 + √ 49 − 24 χ ∗ � χ ( G ) ≤ . 2 Xingxing Yu Coloring
The Four Color Theorem ◮ The Four Color Theorem (Appel and Haken 1977): If G is a planar graph then χ ( G ) ≤ 4. ◮ In 1997, Roberston, Sanders, Seymour, and Thomas gave a simpler and cleaner proof. ◮ Theorem (Ringel and Youngs 1968): If G is embeddable in a surface other than the sphere with Euler characteristic χ ∗ then � 7 + √ 49 − 24 χ ∗ � χ ( G ) ≤ . 2 ◮ Conjecture (Tutte 1966): Every 2-edge-connected graph without Petersen minor has a nowhere-zero 4-flow. Xingxing Yu Coloring
Characterizations of planar graphs ◮ Theorem (Kuratowski 1930): A graph is planar iff it contains no K 3 , 3 -subdivision or K 5 -subdivision. Xingxing Yu Coloring
Characterizations of planar graphs ◮ Theorem (Kuratowski 1930): A graph is planar iff it contains no K 3 , 3 -subdivision or K 5 -subdivision. ◮ Theorem (Wagner 1937): A graph is planar iff it contains no K 3 , 3 -minor or K 5 -minor. Xingxing Yu Coloring
Graphs containing no TK 3 , 3 ◮ Theorem (Wagner 1937): If G contains no K 3 , 3 -subdivision then G is planar, or G ∼ = K 5 , or G admits a cut of size at most 2. Xingxing Yu Coloring
Graphs containing no TK 3 , 3 ◮ Theorem (Wagner 1937): If G contains no K 3 , 3 -subdivision then G is planar, or G ∼ = K 5 , or G admits a cut of size at most 2. ◮ If G does not contain K 3 , 3 -subdivision then χ ( G ) ≤ 5. Xingxing Yu Coloring
Hadwiger’s conjecture ◮ Theorem (Wagner 1937): If G does not contain K 5 -minor and G is edge-maximal then G admits a clique cut of size at most 3, or G is planar, or is the Wagner graph. As a consequence, if G does not contain K 5 -minor then χ ( G ) ≤ 4. Xingxing Yu Coloring
Hadwiger’s conjecture ◮ Theorem (Wagner 1937): If G does not contain K 5 -minor and G is edge-maximal then G admits a clique cut of size at most 3, or G is planar, or is the Wagner graph. As a consequence, if G does not contain K 5 -minor then χ ( G ) ≤ 4. ◮ Conjecture (Hadwiger, 1943): For any k ≥ 1, if G does not contain K k +1 -minor then χ ( G ) ≤ k . Xingxing Yu Coloring
Hadwiger’s conjecture ◮ Theorem (Wagner 1937): If G does not contain K 5 -minor and G is edge-maximal then G admits a clique cut of size at most 3, or G is planar, or is the Wagner graph. As a consequence, if G does not contain K 5 -minor then χ ( G ) ≤ 4. ◮ Conjecture (Hadwiger, 1943): For any k ≥ 1, if G does not contain K k +1 -minor then χ ( G ) ≤ k . ◮ Hadwiger’s conjecture is known to be true for k ≤ 5, but remains open for k ≥ 6. Xingxing Yu Coloring
Hadwiger’s conjecture ◮ Theorem (Wagner 1937): If G does not contain K 5 -minor and G is edge-maximal then G admits a clique cut of size at most 3, or G is planar, or is the Wagner graph. As a consequence, if G does not contain K 5 -minor then χ ( G ) ≤ 4. ◮ Conjecture (Hadwiger, 1943): For any k ≥ 1, if G does not contain K k +1 -minor then χ ( G ) ≤ k . ◮ Hadwiger’s conjecture is known to be true for k ≤ 5, but remains open for k ≥ 6. ◮ The case k = 5 is significantly more complex than the cases for k ≤ 4. Xingxing Yu Coloring
Haj´ os conjecture ◮ Conjecture (Haj´ os 1961): For any integer k ≥ 1, if G does not contain K k +1 -subdivision then χ ( G ) ≤ k . Xingxing Yu Coloring
Haj´ os conjecture ◮ Conjecture (Haj´ os 1961): For any integer k ≥ 1, if G does not contain K k +1 -subdivision then χ ( G ) ≤ k . ◮ Haj´ os’ conjecture is known to be true for k ≤ 3, Xingxing Yu Coloring
Haj´ os conjecture ◮ Conjecture (Haj´ os 1961): For any integer k ≥ 1, if G does not contain K k +1 -subdivision then χ ( G ) ≤ k . ◮ Haj´ os’ conjecture is known to be true for k ≤ 3, ◮ Catlin (1979): Haj´ os’ conjecture is false for every k ≥ 6. Xingxing Yu Coloring
Haj´ os conjecture ◮ Conjecture (Haj´ os 1961): For any integer k ≥ 1, if G does not contain K k +1 -subdivision then χ ( G ) ≤ k . ◮ Haj´ os’ conjecture is known to be true for k ≤ 3, ◮ Catlin (1979): Haj´ os’ conjecture is false for every k ≥ 6. ◮ Erd˝ os and Fajtlowicz (1981): Haj´ os’ conjecture is false for almost all graphs. Xingxing Yu Coloring
Haj´ os conjecture ◮ Conjecture (Haj´ os 1961): For any integer k ≥ 1, if G does not contain K k +1 -subdivision then χ ( G ) ≤ k . ◮ Haj´ os’ conjecture is known to be true for k ≤ 3, ◮ Catlin (1979): Haj´ os’ conjecture is false for every k ≥ 6. ◮ Erd˝ os and Fajtlowicz (1981): Haj´ os’ conjecture is false for almost all graphs. ◮ Thomassen (2005): Haj´ os’ conjecture is false for many obvious reasons. Xingxing Yu Coloring
Fox-Lee-Sudakov conjecture ◮ Let σ ( G ) denotes the largest t such that G contains K t -subdivision, and Xingxing Yu Coloring
Fox-Lee-Sudakov conjecture ◮ Let σ ( G ) denotes the largest t such that G contains K t -subdivision, and ◮ Let H ( n ) := max { χ ( G ) /σ ( G ) : G is a graph with | V ( G ) | = n } . Xingxing Yu Coloring
Fox-Lee-Sudakov conjecture ◮ Let σ ( G ) denotes the largest t such that G contains K t -subdivision, and ◮ Let H ( n ) := max { χ ( G ) /σ ( G ) : G is a graph with | V ( G ) | = n } . ◮ Theorem (Fox, Lee, and Sudakov, 2012): H ( n ) = Θ( √ n / log n ). Xingxing Yu Coloring
Fox-Lee-Sudakov conjecture ◮ Let σ ( G ) denotes the largest t such that G contains K t -subdivision, and ◮ Let H ( n ) := max { χ ( G ) /σ ( G ) : G is a graph with | V ( G ) | = n } . ◮ Theorem (Fox, Lee, and Sudakov, 2012): H ( n ) = Θ( √ n / log n ). ◮ Conjecture (Fox, Lee, and Sudakov, 2012): There is a constant c > 0 such that every graph G with χ ( G ) = k satisfies σ ( G ) ≥ c √ k log k . Xingxing Yu Coloring
Haj´ os conjecture ◮ Theorem (K¨ uhn and Osthus, 2002): Haj´ os’ conjecture holds for large k and graphs with large girth. Xingxing Yu Coloring
Haj´ os conjecture ◮ Theorem (K¨ uhn and Osthus, 2002): Haj´ os’ conjecture holds for large k and graphs with large girth. ◮ Haj´ os’ conjecture remains open for the cases k = 4 and k = 5. Xingxing Yu Coloring
Haj´ os conjecture ◮ Theorem (K¨ uhn and Osthus, 2002): Haj´ os’ conjecture holds for large k and graphs with large girth. ◮ Haj´ os’ conjecture remains open for the cases k = 4 and k = 5. ◮ Find structure of graphs containing no K 5 -subdivision or K 6 -subdivision. Xingxing Yu Coloring
Haj´ os conjecture ◮ Theorem (K¨ uhn and Osthus, 2002): Haj´ os’ conjecture holds for large k and graphs with large girth. ◮ Haj´ os’ conjecture remains open for the cases k = 4 and k = 5. ◮ Find structure of graphs containing no K 5 -subdivision or K 6 -subdivision. ◮ Find structure of a minimum counterexample to Haj´ os’ conjecture. Xingxing Yu Coloring
Graphs containing no TK 5 ◮ Theorem (He, Wang, and Y. 2017): If G does not contain K 5 -subdivision then G is planar or G admits a cut of size at most 4. (This was conjectured independently by Kelmans 1979 and Seymour 1977.) Xingxing Yu Coloring
Graphs containing no TK 5 ◮ Theorem (He, Wang, and Y. 2017): If G does not contain K 5 -subdivision then G is planar or G admits a cut of size at most 4. (This was conjectured independently by Kelmans 1979 and Seymour 1977.) ◮ Corollary. Any counterexample to the Haj´ os conjecture on 4-coloring must have a cut of size at most 4. Xingxing Yu Coloring
Structure of Haj´ os graph A graph G is said to be a Haj´ os graph if (i) G is not 4-colorable, (ii) G contains no K 5 -subdivision, and (iii) subject to (i) and (ii), | V ( G ) | is minimum. Xingxing Yu Coloring
Structure of Haj´ os graph ◮ Theorem (Y. and Zickfeld, 2004). Every Haj´ os graph is 4-connected. G ′ G ′ G 1 G 2 1 2 Xingxing Yu Coloring
Structure of Haj´ os graph ◮ Theorem (Watkins and Mesner 1967): Let G be a 2-connected graph and y 1 , y 2 , y 3 be three distinct vertices in G . Then G does not have a cycle containing { y 1 , y 2 , y 3 } if and only if y 1 y 2 y 3 y 1 y 2 y 3 y 1 y 2 y 3 Xingxing Yu Coloring
Coloring using 4-cuts a x a x a x b y b y b y c z c z c z d d d Xingxing Yu Coloring
K − 4 in Haj´ os graph ◮ Let G be a Haj´ os graph and x 1 , x 2 , y 1 , y 2 ∈ V ( G ) such that G [ { x 1 , x 2 , y 1 , y 2 } ] ∼ = K − 4 and y 1 y 2 / ∈ E ( G ). y 1 x 1 x 2 y 2 Xingxing Yu Coloring
K − 4 in Haj´ os graph y 1 x 1 x 2 y 2 Xingxing Yu Coloring
K − 4 in Haj´ os graph y 1 x 1 x 2 y 2 y 3 Xingxing Yu Coloring
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