Locally identifying colorings of graphs Aline Parreau Joint work - PowerPoint PPT Presentation

Related works With edge colorings : • Vertex-distinguishing edge colorings (Observability of a graph) (Hornak et al, 95’), • Adjacent vertex-distinguishing edge colorings (Zhang et al, 02’) With total colorings : • Adjacent vertex-distinguishing total colorings (Zhang, 05’) 19/40

Link with chromatic number χ lid ( G ) ≥ χ ( G ) Do we need much more than χ ( G ) colors ? 20/40

Link with chromatic number χ lid ( G ) ≥ χ ( G ) Do we need much more than χ ( G ) colors ? 20/40

Link with chromatic number χ lid ( G ) ≥ χ ( G ) Do we need much more than χ ( G ) colors ? An example with χ ( G ) = 3 and χ lid ( G ) ≥ k 20/40

Link with chromatic number χ lid ( G ) ≥ χ ( G ) Do we need much more than χ ( G ) colors ? 1 1 An example with χ ( G ) = 3 and χ lid ( G ) ≥ k 20/40

Link with chromatic number χ lid ( G ) ≥ χ ( G ) Do we need much more than χ ( G ) colors ? 1 2 3 1 An example with χ ( G ) = 3 and χ lid ( G ) ≥ k 20/40

Link with chromatic number χ lid ( G ) ≥ χ ( G ) Do we need much more than χ ( G ) colors ? 1 1 , 2 , 3 ← 2 1 , 2 , 3 ← 3 1 An example with χ ( G ) = 3 and χ lid ( G ) ≥ k 20/40

Link with chromatic number χ lid ( G ) ≥ χ ( G ) Do we need much more than χ ( G ) colors ? An example with χ ( G ) = 3 and χ lid ( G ) ≥ k χ lid is not bounded by a function of χ But... 20/40

Link with maximum degree We have : χ lid ( G ) ≤ χ ( G 3 ) This implies : χ lid ( G ) ≤ ∆( G ) 3 − ∆( G ) 2 + ∆( G ) + 1 21/40

Link with maximum degree We have : χ lid ( G ) ≤ χ ( G 3 ) This implies : χ lid ( G ) ≤ ∆( G ) 3 − ∆( G ) 2 + ∆( G ) + 1 We know only graph that needs ∆( G ) 2 + ∆( G ) + 1 21/40

Transition Slide No bounds with χ for general graphs.. What about “good classes” for proper colorings ? 22/40

Perfect graphs Perfect Line of bipartite Permutation Cograph Chordal Bipartite Bipartite ? Split Interval k -trees Trees Trees ? 23/40

An amazing fact about bipartite graphs G connected graph : • χ lid ( G ) = 1 ⇒ G is a single vertex 24/40

An amazing fact about bipartite graphs G connected graph : • χ lid ( G ) = 1 ⇒ G is a single vertex • χ lid ( G ) = 2 ⇒ G is just an edge 1 2 1 , 2 1 , 2 24/40

An amazing fact about bipartite graphs G connected graph : • χ lid ( G ) = 1 ⇒ G is a single vertex • χ lid ( G ) = 2 ⇒ G is just an edge 1 2 3 1 , 2 1 , 2 24/40

An amazing fact about bipartite graphs G connected graph : • χ lid ( G ) = 1 ⇒ G is a single vertex • χ lid ( G ) = 2 ⇒ G is just an edge 1 2 3 1 , 2 1 , 2 • χ lid ( G ) = 3 ⇒ G is a triangle or a bipartite graph : → Partition vertices with the number of colors they see 24/40

Bipartite graphs L 0 L 1 L 2 L 3 L 4 25/40

Bipartite graphs → L 0 1 1 , 2 → L 1 2 1 , 2 , 3 → L 2 3 2 , 3 , 4 or 2 , 3 → L 3 4 1 , 3 , 4 or 3 , 4 → L 4 1 1 , 4 25/40

Bipartite graphs General bounds : 3 ≤ χ lid ( B ) ≤ 4 26/40

Bipartite graphs General bounds : 3 ≤ χ lid ( B ) ≤ 4 χ lid ( B ) = 3 : 26/40

Bipartite graphs General bounds : 3 ≤ χ lid ( B ) ≤ 4 χ lid ( B ) = 3 : χ lid ( B ) = 4 : 26/40

Bipartite graphs General bounds : 3 ≤ χ lid ( B ) ≤ 4 χ lid ( B ) = 3 : χ lid ( B ) = 4 : ← ? → In general... 3- Lid-Coloring is NP-complete in bipartite graphs 26/40

Link with 2-coloring of hypergraph Try to color a graph with 3 colors 27/40

Link with 2-coloring of hypergraph Try to color a graph with 3 colors 3 1 , 2 , 3 1 2 27/40

Link with 2-coloring of hypergraph Try to color a graph with 3 colors 3 1 , 3 1 , 2 , 3 1 1 , 2 2 1 1 27/40

Link with 2-coloring of hypergraph Try to color a graph with 3 colors 3 1 , 3 1 , 2 , 3 1 1 , 2 2 1 , 2 , 3 1 3 1 , 3 1 , 2 , 3 1 2 1 , 2 1 , 2 , 3 1 2 1 , 2 27/40

Link with 2-coloring of hypergraph Try to color a graph with 3 colors 1 , 3 A 1 , 2 , 3 E 1 , 2 B 1 , 2 , 3 1 , 3 C D A 1 , 2 , 3 1 , 2 D 1 , 2 , 3 C B E 1 , 2 27/40

Link with 2-coloring of hypergraph Try to color a graph with 3 colors 1 , 3 A 1 , 2 , 3 E 1 , 2 B 1 , 2 , 3 1 , 3 C D A 1 , 2 , 3 1 , 2 D 1 , 2 , 3 C B E 1 , 2 • 3- Lid-Coloring in bipartite graph is NP-Complete • Polynomial if B regular, if B is planar with maximum degree 3, if B is a tree. 27/40

Perfect graphs Perfect Line of bipartite Permutation Cograph Chordal Bipartite Bipartite ≤ 4 = 2 ω Split Interval k -trees k -trees ? Trees Trees ≤ 4 = 2 ω 28/40

To perfect graph : k -trees Lid-coloring of 2-trees with 6 colors : • Color the triangle with colors 1 , 2 , 3 • Step : 2 3 i 1 29/40

To perfect graph : k -trees Lid-coloring of 2-trees with 6 colors : • Color the triangle with colors 1 , 2 , 3 • Step : 2 3 i + 3[6] i 1 5 • We always have : ◮ proper coloring ◮ no edge ( i , i + 3) 29/40

To perfect graph : k -trees Lid-coloring of 2-trees with 6 colors : • Color the triangle with colors 1 , 2 , 3 • Step : 2 3 i + 3[6] i 1 5 • We always have : 6 ◮ proper coloring ◮ no edge ( i , i + 3) 29/40

To perfect graph : k -trees Lid-coloring of 2-trees with 6 colors : • Color the triangle with colors 1 , 2 , 3 4 • Step : 2 3 i + 3[6] i 1 5 • We always have : 6 ◮ proper coloring ◮ no edge ( i , i + 3) 29/40

To perfect graph : k -trees Lid-coloring of 2-trees with 6 colors : • Color the triangle with colors 1 , 2 , 3 4 • Step : 2 4 3 i + 3[6] i 1 5 • We always have : 6 ◮ proper coloring ◮ no edge ( i , i + 3) 29/40

To perfect graph : k -trees Lid-coloring of 2-trees with 6 colors : • Color the triangle with colors 1 , 2 , 3 5 6 4 • Step : 2 4 3 i + 3[6] i 1 5 3 3 4 • We always have : 6 2 ◮ proper coloring ◮ no edge ( i , i + 3) 29/40

To perfect graph : k -trees We can extend the construction to k -trees : → A k -tree has lid-chromatic number at most 2 k + 2 This bound is sharp : P k 2 k +2 1 2 3 4 5 6 7 8 30/40

To perfect graph : k -trees We can extend the construction to k -trees : → A k -tree has lid-chromatic number at most 2 k + 2 This bound is sharp : P k 2 k +2 1 1 2 2 3 3 4 4 5 6 7 8 Complete graph 30/40

To perfect graph : k -trees We can extend the construction to k -trees : → A k -tree has lid-chromatic number at most 2 k + 2 This bound is sharp : P k 2 k +2 1 1 2 2 3 3 4 4 5 5 6 7 8 Separated by vertex 5 30/40

To perfect graph : k -trees We can extend the construction to k -trees : → A k -tree has lid-chromatic number at most 2 k + 2 This bound is sharp : P k 2 k +2 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 30/40

Perfect Graphs Perfect Line of bipartite Permutation Cograph Chordal Bipartite Bipartite ≤ 4 = 2 ω Split Interval k -trees k -trees ≤ 2 k + 2 = 2 ω ≤ 4 = 2 ω Trees Trees 31/40

Perfect Graphs Perfect Line of bipartite Permutation Cograph Cograph Chordal ≤ 2 ω − 1 Bipartite Bipartite ≤ 4 = 2 ω Split Split Interval Interval k -trees k -trees ≤ 2 ω ≤ 2 ω − 1 ≤ 2 k + 2 = 2 ω ≤ 4 = 2 ω Trees Trees 31/40

Perfect Graphs Perfect Perfect ? Line of bipartite Permutation Cograph Cograph Chordal Chordal ? ≤ 2 ω − 1 Bipartite Bipartite ≤ 4 = 2 ω Split Split Interval Interval k -trees k -trees ≤ 2 ω ≤ 2 ω − 1 ≤ 2 k + 2 = 2 ω ≤ 4 = 2 ω Trees Trees 31/40

Perfect graphs are not any more perfect... Question : Can we color any perfect graph G with 2 ω ( G ) colors ? 32/40

Perfect graphs are not any more perfect... Question : Can we color any perfect graph G with 2 ω ( G ) colors ? No ! M V 2 V 3 V 1 , V 2 , V 3 stable sets of size k K k , k K k , k \ M V 1 χ lid ≥ k + 2 but ω = 3 32/40

Perfect graphs are not any more perfect... Question : Can we color any perfect graph G with 2 ω ( G ) colors ? No ! M V 2 V 3 V 1 , V 2 , V 3 stable sets of size k K k , k K k , k \ M V 1 χ lid ≥ k + 2 but ω = 3 Conjecture : We can color any chordal graph G with 2 ω ( G ) colors 32/40

Perfect Graphs Perfect Perfect NO Line of bipartite Permutation Cograph Cograph Chordal Chordal ? ≤ 2 ω Bipartite ≤ 2 ω Bipartite Split Split Interval Interval k -trees k -trees ≤ 2 ω ≤ 2 ω ≤ 2 ω ≤ 2 ω Trees Trees 33/40

To support the conjecture : Split graphs Chordal graph : constructed like k -trees but the size of the clique can change 34/40

To support the conjecture : Split graphs Chordal graph : constructed like k -trees but the size of the clique can change Split graphs : K k Independant set 34/40

To support the conjecture : Split graphs Chordal graph : constructed like k -trees but the size of the clique can change Split graphs : • Bondy’s theorem : k − 1 k − 1 vertices of the stable set are enough to separate the clique vertices 34/40

To support the conjecture : Split graphs Chordal graph : constructed like k -trees but the size of the clique can change Split graphs : • Bondy’s theorem : k − 1 k − 1 vertices of the stable set are enough to separate the clique vertices → We can color with 2 k colors → Possible with 2 k − 1 colors 1 → It’s sharp k couleurs 34/40

Planar graphs Is lid-chromatic number bounded for planar graphs ? • Worse example : 8 colors, 35/40