Unique Maximum Facial Colorings Vesna Andova Bernard Lidick y - PowerPoint PPT Presentation

Unique Maximum Facial Colorings Vesna Andova Bernard Lidick´ y Borut Luˇ zar Riste ˇ Kacy Messerschmidt Skrekovski AMS Sectional meeting #1132 Buffalo, NY Sep 16, 2017

Graph Coloring Definition A (proper) coloring of a graph G is a mapping ϕ : V ( G ) → C such that for every uv ∈ E ( G ) : ϕ ( u ) � = ϕ ( v ). G is k-colorable if there is a (proper) coloring of G with | C | = k . Minimum k such that G is k -colorable is denote by χ ( G ). Here we color with { 1 , 2 , . . . , k } instead of arbitrary C . 2

Graph Coloring Definition A (proper) coloring of a graph G is a mapping ϕ : V ( G ) → C such that for every uv ∈ E ( G ) : ϕ ( u ) � = ϕ ( v ). G is k-colorable if there is a (proper) coloring of G with | C | = k . Minimum k such that G is k -colorable is denote by χ ( G ). Here we color with { 1 , 2 , . . . , k } instead of arbitrary C . 2

Plane Graphs Definition A graph G is planar if it can be embeded in the plane, where vertices are points and edges are non-crossing curves. G is plane if it is embedded in the plane. Connected regions of the plane − G are faces . 3

Plane Graphs Definition A graph G is planar if it can be embeded in the plane, where vertices are points and edges are non-crossing curves. G is plane if it is embedded in the plane. Connected regions of the plane − G are faces . 3

Plane Graphs Definition A graph G is planar if it can be embeded in the plane, where vertices are points and edges are non-crossing curves. G is plane if it is embedded in the plane. Connected regions of the plane − G are faces . 3

Theorem (Appel and Haken 1977) Every planar graph is 4 -colorable. 4

Conjecture Conjecture (Fabrici and G¨ oring) If G is a plane graph, then there is a proper coloring of the vertices of G by colors in { 1 , 2 , 3 , 4 } such that every face contains a unique vertex colored with the maximal color appearing on that face. 5

Conjecture Conjecture (Fabrici and G¨ oring) If G is a plane graph, then there is a proper coloring of the vertices of G by colors in { 1 , 2 , 3 , 4 } such that every face contains a unique vertex colored with the maximal color appearing on that face. 5

Conjecture Conjecture (Fabrici and G¨ oring) If G is a plane graph, then there is a proper coloring of the vertices of G by colors in { 1 , 2 , 3 , 4 } such that every face contains a unique vertex colored with the maximal color appearing on that face. 2 1 4 1 4 3 5

Conjecture Conjecture (Fabrici and G¨ oring) If G is a plane graph, then there is a proper coloring of the vertices of G by colors in { 1 , 2 , 3 , 4 } such that every face contains a unique vertex colored with the maximal color appearing on that face. 2 1 4 1 4 3 5

Conjecture Conjecture (Fabrici and G¨ oring) If G is a plane graph, then there is a proper coloring of the vertices of G by colors in { 1 , 2 , 3 , 4 } such that every face contains a unique vertex colored with the maximal color appearing on that face. 2 1 4 1 4 3 Note: Add or delete edges carefully! 5

Conjecture Conjecture (Fabrici and G¨ oring) If G is a plane graph, then χ fum ( G ) ≤ 4 . A proper coloring of a graph G embedded on some surface, where (1) colors are natural numbers, and (2) every face has a unique vertex colored with its maximal color, is called a facial unique-maximum coloring or FUM-coloring . The minimum number k such that G admits a FUM-coloring with colors { 1 , 2 , . . . , k } is called the FUM chromatic number of G , denoted by χ fum ( G ). 6

Conjecture Conjecture (Fabrici and G¨ oring) If G is a plane graph, then χ fum ( G ) ≤ 4 . 7

Conjecture Conjecture (Fabrici and G¨ oring) If G is a plane graph, then χ fum ( G ) ≤ 4 . Theorem (Fabrici and G¨ oring 2015) If G is a plane graph, then χ fum ( G ) ≤ 6 . 7

Conjecture Conjecture (Fabrici and G¨ oring) If G is a plane graph, then χ fum ( G ) ≤ 4 . Theorem (Fabrici and G¨ oring 2015) If G is a plane graph, then χ fum ( G ) ≤ 6 . Theorem (Wendland 2016) If G is a plane graph, then χ fum ( G ) ≤ 5 . 7

Idea Theorem (Fabrici and G¨ oring 2015) If G is a plane graph, then χ fum ( G ) ≤ 6 . 8

Idea Theorem (Fabrici and G¨ oring 2015) If G is a plane graph, then χ fum ( G ) ≤ 6 . Color some vertices of G by colors 5 and 6 such that each face contains unique 6 or (no 6 and unique 5). 6 6 5 6 5 8

Idea Theorem (Fabrici and G¨ oring 2015) If G is a plane graph, then χ fum ( G ) ≤ 6 . Color some vertices of G by colors 5 and 6 such that each face contains unique 6 or (no 6 and unique 5). 6 6 5 6 5 Color rest by 4-color theorem with { 1 , 2 , 3 , 4 } . 8

Idea Theorem (Fabrici and G¨ oring 2015) If G is a plane graph, then χ fum ( G ) ≤ 6 . Color some vertices of G by colors 5 and 6 such that each face contains unique 6 or (no 6 and unique 5). 4 5 Color rest by 4-color theorem with { 1 , 2 , 3 , 4 } . Wendland: Make the rest triangle-free and use Gr¨ otzsch’s theorem. Just { 4 , 5 } ∪ { 1 , 2 , 3 } colors needed in total. 8

Our Results Conjecture (Fabrici and G¨ oring) If G is a plane graph, then χ fum ( G ) ≤ 4 . 9

Our Results Conjecture (Fabrici and G¨ oring) If G is a plane graph, then χ fum ( G ) ≤ 4 . zar, ˇ Theorem (Andova, L., Luˇ Skrekovski) If G is a plane subcubic graph, then χ fum ( G ) ≤ 4 . 9

Our Results Conjecture (Fabrici and G¨ oring) If G is a plane graph, then χ fum ( G ) ≤ 4 . zar, ˇ Theorem (Andova, L., Luˇ Skrekovski) If G is a plane subcubic graph, then χ fum ( G ) ≤ 4 . zar, ˇ Theorem (Andova, L., Luˇ Skrekovski) If G is an outerplane graph, then χ fum ( G ) ≤ 4 . Both results are tight. 9

Tight Example For the following graph G , χ fum ( G ) > 3. Suppose for contradiction χ fum ( G ) = 3: Notice G is subcubic, bipartite, 2-connected, and outerplane. 10

Tight Example For the following graph G , χ fum ( G ) > 3. Suppose for contradiction χ fum ( G ) = 3: Notice G is subcubic, bipartite, 2-connected, and outerplane. 10

Tight Example For the following graph G , χ fum ( G ) > 3. Suppose for contradiction χ fum ( G ) = 3: 1 2 1 2 1 2 Notice G is subcubic, bipartite, 2-connected, and outerplane. 10

Tight Example For the following graph G , χ fum ( G ) > 3. Suppose for contradiction χ fum ( G ) = 3: 3 Notice G is subcubic, bipartite, 2-connected, and outerplane. 10

Tight Example For the following graph G , χ fum ( G ) > 3. Suppose for contradiction χ fum ( G ) = 3: 3 2 1 2 1 2 2 1 1 2 2 1 1 2 1 2 1 2 Notice G is subcubic, bipartite, 2-connected, and outerplane. 10

Tight Example For the following graph G , χ fum ( G ) > 3. Suppose for contradiction χ fum ( G ) = 3: 3 2 1 2 1 2 2 1 1 ! 2 2 1 1 2 1 2 1 2 Notice G is subcubic, bipartite, 2-connected, and outerplane. Also, G can have arbitrarily large girth. 10

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Proof Idea If G is a plane subcubic graph, then χ fum ( G ) ≤ 4. Use precoloring extension method (Thomassen’s 5-list coloring) (Nice induction) 12

Proof Idea If G is a plane subcubic graph, then χ fum ( G ) ≤ 4. Use precoloring extension method (Thomassen’s 5-list coloring) (Nice induction) 4 12

Proof Idea If G is a plane subcubic graph, then χ fum ( G ) ≤ 4. Use precoloring extension method (Thomassen’s 5-list coloring) (Nice induction) 4 outer face colored by { 1 , 2 , 3 } 12

Proof Idea If G is a plane subcubic graph, then χ fum ( G ) ≤ 4. Use precoloring extension method (Thomassen’s 5-list coloring) (Nice induction) 2 1 4 outer face colored by { 1 , 2 , 3 } up to two precolored vertices (eliminate cut vertices and chords) 12

Proof Idea If G is a plane subcubic graph, then χ fum ( G ) ≤ 4. Use precoloring extension method (Thomassen’s 5-list coloring) (Nice induction) 2 1 4 outer face colored by { 1 , 2 , 3 } up to two precolored vertices (eliminate cut vertices and chords) 12

Proof Idea If G is a plane subcubic graph, then χ fum ( G ) ≤ 4. Use precoloring extension method (Thomassen’s 5-list coloring) (Nice induction) 2 1 4 outer face colored by { 1 , 2 , 3 } up to two precolored vertices (eliminate cut vertices and chords) 12

Proof Idea If G is a plane subcubic graph, then χ fum ( G ) ≤ 4. Use precoloring extension method (Thomassen’s 5-list coloring) (Nice induction) 3 2 1 1 3 2 4 outer face colored by { 1 , 2 , 3 } up to two precolored vertices (eliminate cut vertices and chords) 12

Proof Idea If G is a plane subcubic graph, then χ fum ( G ) ≤ 4. Use precoloring extension method (Thomassen’s 5-list coloring) (Nice induction) 3 2 1 1 3 2 4 outer face colored by { 1 , 2 , 3 } up to two precolored vertices (eliminate cut vertices and chords) 12

Proof Idea If G is a plane subcubic graph, then χ fum ( G ) ≤ 4. Use precoloring extension method (Thomassen’s 5-list coloring) (Nice induction) 3 2 2 4 1 1 3 2 4 outer face colored by { 1 , 2 , 3 } up to two precolored vertices (eliminate cut vertices and chords) 12

Proof Idea If G is a plane subcubic graph, then χ fum ( G ) ≤ 4. Use precoloring extension method (Thomassen’s 5-list coloring) (Nice induction) 3 2 2 4 1 1 3 2 4 outer face colored by { 1 , 2 , 3 } up to two precolored vertices (eliminate cut vertices and chords) 12