Strong edge-colorings of sparse graphs with large maximum degree - PowerPoint PPT Presentation

Strong edge-colorings of sparse graphs with large maximum degree Strong edge-colorings of sparse graphs with large maximum degree ILKYOO CHOI KAIST, Korea Joint work with Jaehoon Kim Alexandr Kostochka Andr´ e Raspaud

Strong edge-colorings of sparse graphs with large maximum degree A proper (vertex) coloring : partition V ( G ) into independent sets

Strong edge-colorings of sparse graphs with large maximum degree A proper (vertex) coloring : partition V ( G ) into independent sets – Greedy bound: χ ( G ) ≤ ∆( G ) + 1

Strong edge-colorings of sparse graphs with large maximum degree A proper (vertex) coloring : partition V ( G ) into independent sets – Greedy bound: χ ( G ) ≤ ∆( G ) + 1 · · ·

Strong edge-colorings of sparse graphs with large maximum degree A proper (vertex) coloring : partition V ( G ) into independent sets – Greedy bound: χ ( G ) ≤ ∆( G ) + 1 · · ·

Strong edge-colorings of sparse graphs with large maximum degree A proper (vertex) coloring : partition V ( G ) into independent sets – Greedy bound: χ ( G ) ≤ ∆( G ) + 1 · · ·

Strong edge-colorings of sparse graphs with large maximum degree A proper (vertex) coloring : partition V ( G ) into independent sets – Greedy bound: χ ( G ) ≤ ∆( G ) + 1

Strong edge-colorings of sparse graphs with large maximum degree A proper (vertex) coloring : partition V ( G ) into independent sets – Greedy bound: χ ( G ) ≤ ∆( G ) + 1 Theorem (1941 Brooks) If G is neither a complete graph nor an odd cycle, then χ ( G ) ≤ ∆( G ) .

Strong edge-colorings of sparse graphs with large maximum degree A proper (vertex) coloring : partition V ( G ) into independent sets – Greedy bound: χ ( G ) ≤ ∆( G ) + 1 Theorem (1941 Brooks) If G is neither a complete graph nor an odd cycle, then χ ( G ) ≤ ∆( G ) . A proper edge-coloring : partition E ( G ) into matchings

Strong edge-colorings of sparse graphs with large maximum degree A proper (vertex) coloring : partition V ( G ) into independent sets – Greedy bound: χ ( G ) ≤ ∆( G ) + 1 Theorem (1941 Brooks) If G is neither a complete graph nor an odd cycle, then χ ( G ) ≤ ∆( G ) . A proper edge-coloring : partition E ( G ) into matchings – Greedy bound: χ ′ ( G ) ≤ 2(∆( G ) − 1) + 1

Strong edge-colorings of sparse graphs with large maximum degree A proper (vertex) coloring : partition V ( G ) into independent sets – Greedy bound: χ ( G ) ≤ ∆( G ) + 1 Theorem (1941 Brooks) If G is neither a complete graph nor an odd cycle, then χ ( G ) ≤ ∆( G ) . A proper edge-coloring : partition E ( G ) into matchings – Greedy bound: χ ′ ( G ) ≤ 2(∆( G ) − 1) + 1

Strong edge-colorings of sparse graphs with large maximum degree A proper (vertex) coloring : partition V ( G ) into independent sets – Greedy bound: χ ( G ) ≤ ∆( G ) + 1 Theorem (1941 Brooks) If G is neither a complete graph nor an odd cycle, then χ ( G ) ≤ ∆( G ) . A proper edge-coloring : partition E ( G ) into matchings – Greedy bound: χ ′ ( G ) ≤ 2(∆( G ) − 1) + 1

Strong edge-colorings of sparse graphs with large maximum degree A proper (vertex) coloring : partition V ( G ) into independent sets – Greedy bound: χ ( G ) ≤ ∆( G ) + 1 Theorem (1941 Brooks) If G is neither a complete graph nor an odd cycle, then χ ( G ) ≤ ∆( G ) . A proper edge-coloring : partition E ( G ) into matchings – Greedy bound: χ ′ ( G ) ≤ 2(∆( G ) − 1) + 1 Theorem (1976 Vizing) For a graph G, χ ′ ( G ) ≤ ∆( G ) + 1 .

Strong edge-colorings of sparse graphs with large maximum degree A proper (vertex) coloring : partition V ( G ) into independent sets – Greedy bound: χ ( G ) ≤ ∆( G ) + 1 Theorem (1941 Brooks) If G is neither a complete graph nor an odd cycle, then χ ( G ) ≤ ∆( G ) . A proper edge-coloring : partition E ( G ) into matchings – Greedy bound: χ ′ ( G ) ≤ 2(∆( G ) − 1) + 1 Theorem (1976 Vizing) For a graph G, χ ′ ( G ) ≤ ∆( G ) + 1 . A strong edge-coloring : partition E ( G ) into induced matchings

Strong edge-colorings of sparse graphs with large maximum degree A proper (vertex) coloring : partition V ( G ) into independent sets – Greedy bound: χ ( G ) ≤ ∆( G ) + 1 Theorem (1941 Brooks) If G is neither a complete graph nor an odd cycle, then χ ( G ) ≤ ∆( G ) . A proper edge-coloring : partition E ( G ) into matchings – Greedy bound: χ ′ ( G ) ≤ 2(∆( G ) − 1) + 1 Theorem (1976 Vizing) For a graph G, χ ′ ( G ) ≤ ∆( G ) + 1 . A strong edge-coloring : partition E ( G ) into induced matchings – Greedy bound: χ ′ s ( G ) ≤ 2∆( G )(∆( G ) − 1) + 1

Strong edge-colorings of sparse graphs with large maximum degree A proper (vertex) coloring : partition V ( G ) into independent sets – Greedy bound: χ ( G ) ≤ ∆( G ) + 1 Theorem (1941 Brooks) If G is neither a complete graph nor an odd cycle, then χ ( G ) ≤ ∆( G ) . A proper edge-coloring : partition E ( G ) into matchings – Greedy bound: χ ′ ( G ) ≤ 2(∆( G ) − 1) + 1 Theorem (1976 Vizing) For a graph G, χ ′ ( G ) ≤ ∆( G ) + 1 . A strong edge-coloring : partition E ( G ) into induced matchings – Greedy bound: χ ′ s ( G ) ≤ 2∆( G )(∆( G ) − 1) + 1

Strong edge-colorings of sparse graphs with large maximum degree A proper (vertex) coloring : partition V ( G ) into independent sets – Greedy bound: χ ( G ) ≤ ∆( G ) + 1 Theorem (1941 Brooks) If G is neither a complete graph nor an odd cycle, then χ ( G ) ≤ ∆( G ) . A proper edge-coloring : partition E ( G ) into matchings – Greedy bound: χ ′ ( G ) ≤ 2(∆( G ) − 1) + 1 Theorem (1976 Vizing) For a graph G, χ ′ ( G ) ≤ ∆( G ) + 1 . A strong edge-coloring : partition E ( G ) into induced matchings – Greedy bound: χ ′ s ( G ) ≤ 2∆( G )(∆( G ) − 1) + 1

Strong edge-colorings of sparse graphs with large maximum degree A proper (vertex) coloring : partition V ( G ) into independent sets – Greedy bound: χ ( G ) ≤ ∆( G ) + 1 Theorem (1941 Brooks) If G is neither a complete graph nor an odd cycle, then χ ( G ) ≤ ∆( G ) . A proper edge-coloring : partition E ( G ) into matchings – Greedy bound: χ ′ ( G ) ≤ 2(∆( G ) − 1) + 1 Theorem (1976 Vizing) For a graph G, χ ′ ( G ) ≤ ∆( G ) + 1 . A strong edge-coloring : partition E ( G ) into induced matchings – Greedy bound: χ ′ s ( G ) ≤ 2∆( G )(∆( G ) − 1) + 1 Conjecture (1989 Erd˝ os, Neˇ setˇ ril) � 1 . 25∆( G ) 2 ∆( G ) is even For a graph G, χ ′ s ( G ) ≤ 1 . 25∆( G ) 2 − 0 . 5∆( G ) + 0 . 25 ∆( G ) is odd

Strong edge-colorings of sparse graphs with large maximum degree A strong edge-coloring : partition E ( G ) into induced matchings Greedy bound: χ ′ s ( G ) ≤ 2∆( G )(∆( G ) − 1) + 1 Conjecture (1989 Erd˝ os, Neˇ setˇ ril) � 1 . 25∆( G ) 2 ∆( G ) is even For a graph G, χ ′ s ( G ) ≤ 1 . 25∆( G ) 2 − 0 . 5∆( G ) + 0 . 25 ∆( G ) is odd

Strong edge-colorings of sparse graphs with large maximum degree A strong edge-coloring : partition E ( G ) into induced matchings Greedy bound: χ ′ s ( G ) ≤ 2∆( G )(∆( G ) − 1) + 1 Conjecture (1989 Erd˝ os, Neˇ setˇ ril) � 1 . 25∆( G ) 2 ∆( G ) is even For a graph G, χ ′ s ( G ) ≤ 1 . 25∆( G ) 2 − 0 . 5∆( G ) + 0 . 25 ∆( G ) is odd If true, then sharp: blowup of a 5-cycle.

Strong edge-colorings of sparse graphs with large maximum degree A strong edge-coloring : partition E ( G ) into induced matchings Greedy bound: χ ′ s ( G ) ≤ 2∆( G )(∆( G ) − 1) + 1 Conjecture (1989 Erd˝ os, Neˇ setˇ ril) � 1 . 25∆( G ) 2 ∆( G ) is even For a graph G, χ ′ s ( G ) ≤ 1 . 25∆( G ) 2 − 0 . 5∆( G ) + 0 . 25 ∆( G ) is odd If true, then sharp: blowup of a 5-cycle. ∆ ∆ − 1 2 2 ∆ ∆ ∆+1 ∆+1 2 2 2 2 ∆ ∆ ∆ − 1 ∆ − 1 2 2 2 2

Strong edge-colorings of sparse graphs with large maximum degree A strong edge-coloring : partition E ( G ) into induced matchings Greedy bound: χ ′ s ( G ) ≤ 2∆( G )(∆( G ) − 1) + 1 Conjecture (1989 Erd˝ os, Neˇ setˇ ril) � 1 . 25∆( G ) 2 ∆( G ) is even For a graph G, χ ′ s ( G ) ≤ 1 . 25∆( G ) 2 − 0 . 5∆( G ) + 0 . 25 ∆( G ) is odd If true, then sharp: blowup of a 5-cycle.

Strong edge-colorings of sparse graphs with large maximum degree A strong edge-coloring : partition E ( G ) into induced matchings Greedy bound: χ ′ s ( G ) ≤ 2∆( G )(∆( G ) − 1) + 1 Conjecture (1989 Erd˝ os, Neˇ setˇ ril) � 1 . 25∆( G ) 2 ∆( G ) is even For a graph G, χ ′ s ( G ) ≤ 1 . 25∆( G ) 2 − 0 . 5∆( G ) + 0 . 25 ∆( G ) is odd If true, then sharp: blowup of a 5-cycle. Exact results:

Strong edge-colorings of sparse graphs with large maximum degree A strong edge-coloring : partition E ( G ) into induced matchings Greedy bound: χ ′ s ( G ) ≤ 2∆( G )(∆( G ) − 1) + 1 Conjecture (1989 Erd˝ os, Neˇ setˇ ril) � 1 . 25∆( G ) 2 ∆( G ) is even For a graph G, χ ′ s ( G ) ≤ 1 . 25∆( G ) 2 − 0 . 5∆( G ) + 0 . 25 ∆( G ) is odd If true, then sharp: blowup of a 5-cycle. Exact results: Theorem (1992 Anderson, 1993 Hor´ ak, Qing, Trotter) If ∆( G ) = 3 , then χ ′ s ( G ) ≤ 10 .