Strong chromatic index of graphs with maximum degree four
Strong chromatic index of graphs with maximum degree four Michael Santana
Joint Work with M. Huang and G. Yu May 2017
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Definition Definition Given a graph G, a strong edge-coloring is a coloring of E(G) such that every color class forms an induced matching in G.
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Definition Definition Given a graph G, a strong edge-coloring is a coloring of E(G) such that every color class forms an induced matching in G.
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Definition Definition Given a graph G, a strong edge-coloring is a coloring of E(G) such that every color class forms an induced matching in G. Definition The strong chromatic index of G, denoted by χs0 (G), is the minimum number of colors needed for a strong edge-coloring of G.
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Bounds Proposition For every graph G with maximum degree ∆, ∆ ≤ χ0 (G) ≤ χs0 (G)
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Bounds Proposition For every graph G with maximum degree ∆, ∆ ≤ χ0 (G) ≤ χs0 (G) ≤ 2∆(∆ − 1) + 1.
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Bounds Proposition For every graph G with maximum degree ∆, ∆ ≤ χs0 (G) ≤ 2∆2 .
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Bounds Proposition For every graph G with maximum degree ∆, ∆ ≤ χs0 (G) ≤ 2∆2 . The lower bound is best possible due to K1,∆ .
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Bounds Proposition For every graph G with maximum degree ∆, ∆ ≤ χs0 (G) ≤ 2∆2 . The lower bound is best possible due to K1,∆ . The order of magnitude of the upper bound is also best possible as ∆+1 1 0 χs (K∆+1 ) = ≈ ∆2 . 2 2
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Conjecture Conjecture (Erd˝ os-Neˇ setˇril ‘85) For any ¨ graph G with maximum degree ∆, 5 2 ∆ , for even ∆ 0 4 χs (G) ≤ 5 2 1 1 ∆ − 2 ∆ + 4 , for odd ∆ 4
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Blow-Up of C5
c b∆ 2 b∆ c 2
b∆ c 2
d∆ e 2
d∆ e 2 5 / 16
Blow-Up of C5
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Blow-Up of C5
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Blow-Up of C5
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Blow-Up of C5
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Blow-Up of C5
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Blow-Up of C5
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Blow-Up of C5
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Blow-Up of C5
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Blow-Up of C5 ¨5
χs0 (Blow-up
of C5 ) =
∆2 , 4 5 2 ∆ − 12 ∆ + 14 , 4
for even ∆ for odd ∆
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Blow-Up of C5 ¨5
χs0 (Blow-up
of C5 ) =
∆2 , 4 5 2 ∆ − 12 ∆ + 14 , 4
for even ∆ for odd ∆
If G is (2K2 )-free, then χs0 (G) = |E(G)|.
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Blow-Up of C5 ¨5
χs0 (Blow-up
of C5 ) =
∆2 , 4 5 2 ∆ − 12 ∆ + 14 , 4
for even ∆ for odd ∆
If G is (2K2 )-free, then χs0 (G) = |E(G)|. Theorem (Chung-Gyárfás-Trotter-Tuza ‘90) The number of edges¨ in a (2K2 )-free graph with max 5 2 ∆ , for even ∆ degree ∆ is at most 45 2 1 1 ∆ − 2 ∆ + 4 , for odd ∆. 4 Additionally, the blow-up of C5 is the unique extremal graph.
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Conjecture Conjecture (Erd˝ os-Neˇ setˇril ‘85) For any ¨ graph G with maximum degree ∆, 5 2 ∆ , for even ∆ 0 4 χs (G) ≤ 5 2 1 1 ∆ − 2 ∆ + 4 , for odd ∆ 4
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Conjecture Conjecture (Erd˝ os-Neˇ setˇril ‘85) For any ¨ graph G with maximum degree ∆, 5 2 ∆ , for even ∆ 0 4 χs (G) ≤ 5 2 1 1 ∆ − 2 ∆ + 4 , for odd ∆ 4 χs0 (G) ≤ 1.998∆2 (Molloy-Reed ‘97)
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Conjecture Conjecture (Erd˝ os-Neˇ setˇril ‘85) For any ¨ graph G with maximum degree ∆, 5 2 ∆ , for even ∆ 0 4 χs (G) ≤ 5 2 1 1 ∆ − 2 ∆ + 4 , for odd ∆ 4 χs0 (G) ≤ 1.998∆2 (Molloy-Reed ‘97) χs0 (G) ≤ 1.93∆2 (Bruhn-Joos ‘15)
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Conjecture Conjecture (Erd˝ os-Neˇ setˇril ‘85) For any ¨ graph G with maximum degree ∆, 5 2 ∆ , for even ∆ 0 4 χs (G) ≤ 5 2 1 1 ∆ − 2 ∆ + 4 , for odd ∆ 4 χs0 (G) ≤ 1.998∆2 (Molloy-Reed ‘97) χs0 (G) ≤ 1.93∆2 (Bruhn-Joos ‘15) χs0 (G) ≤ 1.835∆2 (Bonamy-Perrett-Postle ‘17)
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Conjecture Conjecture (Erd˝ os-Neˇ setˇril ‘85) For any ¨ graph G with maximum degree ∆, 5 2 ∆ , for even ∆ 0 4 χs (G) ≤ 5 2 1 1 ∆ − 2 ∆ + 4 , for odd ∆ 4 χs0 (G) ≤ 1.998∆2 (Molloy-Reed ‘97) χs0 (G) ≤ 1.93∆2 (Bruhn-Joos ‘15) χs0 (G) ≤ 1.835∆2 (Bonamy-Perrett-Postle ‘17) Proven for ∆ = 3 (Andersen ‘92, Horák-Qing-Trotter ‘93)
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Conjecture Conjecture (Erd˝ os-Neˇ setˇril ‘85) For any ¨ graph G with maximum degree ∆, 5 2 ∆ , for even ∆ 0 4 χs (G) ≤ 5 2 1 1 ∆ − 2 ∆ + 4 , for odd ∆ 4 χs0 (G) ≤ 1.998∆2 (Molloy-Reed ‘97) χs0 (G) ≤ 1.93∆2 (Bruhn-Joos ‘15) χs0 (G) ≤ 1.835∆2 (Bonamy-Perrett-Postle ‘17) Proven for ∆ = 3 (Andersen ‘92, Horák-Qing-Trotter ‘93) For ∆ = 4, χs0 (G) ≤ 22 (Cranston ‘06)
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Conjecture Conjecture (Erd˝ os-Neˇ setˇril ‘85) For any ¨ graph G with maximum degree ∆, 5 2 ∆ , for even ∆ 0 4 χs (G) ≤ 5 2 1 1 ∆ − 2 ∆ + 4 , for odd ∆ 4 χs0 (G) ≤ 1.998∆2 (Molloy-Reed ‘97) χs0 (G) ≤ 1.93∆2 (Bruhn-Joos ‘15) χs0 (G) ≤ 1.835∆2 (Bonamy-Perrett-Postle ‘17) Proven for ∆ = 3 (Andersen ‘92, Horák-Qing-Trotter ‘93) For ∆ = 4, χs0 (G) ≤ 22 (Cranston ‘06) For ∆ = 4, χs0 (G) ≤ 21 (Huang-S-Yu ‘17++)
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Proof Sketch Theorem (Huang-S-Yu ‘17++) If G is a multigraph with ∆(G) ≤ 4, then χs0 (G) ≤ 21.
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Proof Sketch Theorem (Huang-S-Yu ‘17++) If G is a multigraph with ∆(G) ≤ 4, then χs0 (G) ≤ 21. Among all counterexamples, choose G so that |V(G)| + |E(G)| is minimized.
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Proof Sketch Theorem (Huang-S-Yu ‘17++) If G is a multigraph with ∆(G) ≤ 4, then χs0 (G) ≤ 21. Among all counterexamples, choose G so that |V(G)| + |E(G)| is minimized. So ∆(G) ≤ 4 and χs0 (G) > 21.
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Proof Sketch Properties of a Minimal Counterexample G G is 4-regular, simple, etc.
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Proof Sketch Properties of a Minimal Counterexample G G is 4-regular, simple, etc. G has girth at least 6.
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Proof Sketch Properties of a Minimal Counterexample G G is 4-regular, simple, etc. G has girth at least 6. G has no edge-cut of size at most 3.
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Proof Sketch Properties of a Minimal Counterexample G G is 4-regular, simple, etc. G has girth at least 6. G has no edge-cut of size at most 3. How to Color the Edges of G
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Proof Sketch Properties of a Minimal Counterexample G G is 4-regular, simple, etc. G has girth at least 6. G has no edge-cut of size at most 3. How to Color the Edges of G Partition the vertices of G into three sets (L, M, and R), where M is a cut-set
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Proof Sketch Properties of a Minimal Counterexample G G is 4-regular, simple, etc. G has girth at least 6. G has no edge-cut of size at most 3. How to Color the Edges of G Partition the vertices of G into three sets (L, M, and R), where M is a cut-set Show that M contains some special vertices.
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Proof Sketch Properties of a Minimal Counterexample G G is 4-regular, simple, etc. G has girth at least 6. G has no edge-cut of size at most 3. How to Color the Edges of G Partition the vertices of G into three sets (L, M, and R), where M is a cut-set Show that M contains some special vertices. Case analysis and color.
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Open Problems Conjecture (Erd˝ os-Neˇ setˇril ‘85) If ∆(G) ≤ 4, then χs0 (G) ≤ 20.
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Open Problems Conjecture (Erd˝ os-Neˇ setˇril ‘85) If ∆(G) ≤ 4, then χs0 (G) ≤ 20. Conjecture (Faudree-Gyárfás-Schelp-Tuza ‘90) Suppose G is a bipartite graph with maximum degree ∆.
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Open Problems Conjecture (Erd˝ os-Neˇ setˇril ‘85) If ∆(G) ≤ 4, then χs0 (G) ≤ 20. Conjecture (Faudree-Gyárfás-Schelp-Tuza ‘90) Suppose G is a bipartite graph with maximum degree ∆. 1
χs0 (G) ≤ ∆2 .
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Open Problems Conjecture (Erd˝ os-Neˇ setˇril ‘85) If ∆(G) ≤ 4, then χs0 (G) ≤ 20. Conjecture (Faudree-Gyárfás-Schelp-Tuza ‘90) Suppose G is a bipartite graph with maximum degree ∆. 1
χs0 (G) ≤ ∆2 .
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If ∆ ≤ 3 and G has girth at least six, then χs0 (G) ≤ 7
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Open Problems Conjecture (Erd˝ os-Neˇ setˇril ‘85) If ∆(G) ≤ 4, then χs0 (G) ≤ 20. Conjecture (Faudree-Gyárfás-Schelp-Tuza ‘90) Suppose G is a bipartite graph with maximum degree ∆. 1
χs0 (G) ≤ ∆2 .
2
If ∆ ≤ 3 and G has girth at least six, then χs0 (G) ≤ 7
3
If ∆ ≤ 3 and G has ‘large’ girth, then χs0 (G) ≤ 5
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Open Problems Conjecture (Erd˝ os-Neˇ setˇril ‘85) If ∆(G) ≤ 4, then χs0 (G) ≤ 20. Conjecture (Faudree-Gyárfás-Schelp-Tuza ‘90) Suppose G is a bipartite graph with maximum degree ∆. 1
χs0 (G) ≤ ∆2 .
2
If ∆ ≤ 3 and G has girth at least six, then χs0 (G) ≤ 7
3
If ∆ ≤ 3 and G has ‘large’ girth, then χs0 (G) ≤ 5
Theorem (Faudree et al. ‘90) If G is a planar graph with maximum degree ∆, then 4∆ − 4 ≤ χs0 (G) ≤ 4∆ + 4. 13 / 16
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MIGHTY LVIII Grand Valley State University October 6-7, 2017 Plenary Speakers: Doug West David Galvin www.gvsu.edu/math/mighty-lviii MIGHTY_LVIII@ gvsu.edu
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Thanks for your attention!
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Strong chromatic index of graphs with maximum degree four Michael Santana
Joint Work with M. Huang and G. Yu May 2017
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Joint Work with M. Huang and G. Yu May 2017
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Definition Definition Given a graph G, a strong edge-coloring is a coloring of E(G) such that every color class forms an induced matching in G.
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Definition Definition Given a graph G, a strong edge-coloring is a coloring of E(G) such that every color class forms an induced matching in G.
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Definition Definition Given a graph G, a strong edge-coloring is a coloring of E(G) such that every color class forms an induced matching in G. Definition The strong chromatic index of G, denoted by χs0 (G), is the minimum number of colors needed for a strong edge-coloring of G.
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Bounds Proposition For every graph G with maximum degree ∆, ∆ ≤ χ0 (G) ≤ χs0 (G)
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Bounds Proposition For every graph G with maximum degree ∆, ∆ ≤ χ0 (G) ≤ χs0 (G) ≤ 2∆(∆ − 1) + 1.
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Bounds Proposition For every graph G with maximum degree ∆, ∆ ≤ χs0 (G) ≤ 2∆2 .
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Bounds Proposition For every graph G with maximum degree ∆, ∆ ≤ χs0 (G) ≤ 2∆2 . The lower bound is best possible due to K1,∆ .
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Bounds Proposition For every graph G with maximum degree ∆, ∆ ≤ χs0 (G) ≤ 2∆2 . The lower bound is best possible due to K1,∆ . The order of magnitude of the upper bound is also best possible as ∆+1 1 0 χs (K∆+1 ) = ≈ ∆2 . 2 2
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Conjecture Conjecture (Erd˝ os-Neˇ setˇril ‘85) For any ¨ graph G with maximum degree ∆, 5 2 ∆ , for even ∆ 0 4 χs (G) ≤ 5 2 1 1 ∆ − 2 ∆ + 4 , for odd ∆ 4
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Blow-Up of C5
c b∆ 2 b∆ c 2
b∆ c 2
d∆ e 2
d∆ e 2 5 / 16
Blow-Up of C5
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Blow-Up of C5
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Blow-Up of C5
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Blow-Up of C5
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Blow-Up of C5
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Blow-Up of C5
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Blow-Up of C5
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Blow-Up of C5
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Blow-Up of C5 ¨5
χs0 (Blow-up
of C5 ) =
∆2 , 4 5 2 ∆ − 12 ∆ + 14 , 4
for even ∆ for odd ∆
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Blow-Up of C5 ¨5
χs0 (Blow-up
of C5 ) =
∆2 , 4 5 2 ∆ − 12 ∆ + 14 , 4
for even ∆ for odd ∆
If G is (2K2 )-free, then χs0 (G) = |E(G)|.
7 / 16
Blow-Up of C5 ¨5
χs0 (Blow-up
of C5 ) =
∆2 , 4 5 2 ∆ − 12 ∆ + 14 , 4
for even ∆ for odd ∆
If G is (2K2 )-free, then χs0 (G) = |E(G)|. Theorem (Chung-Gyárfás-Trotter-Tuza ‘90) The number of edges¨ in a (2K2 )-free graph with max 5 2 ∆ , for even ∆ degree ∆ is at most 45 2 1 1 ∆ − 2 ∆ + 4 , for odd ∆. 4 Additionally, the blow-up of C5 is the unique extremal graph.
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Conjecture Conjecture (Erd˝ os-Neˇ setˇril ‘85) For any ¨ graph G with maximum degree ∆, 5 2 ∆ , for even ∆ 0 4 χs (G) ≤ 5 2 1 1 ∆ − 2 ∆ + 4 , for odd ∆ 4
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Conjecture Conjecture (Erd˝ os-Neˇ setˇril ‘85) For any ¨ graph G with maximum degree ∆, 5 2 ∆ , for even ∆ 0 4 χs (G) ≤ 5 2 1 1 ∆ − 2 ∆ + 4 , for odd ∆ 4 χs0 (G) ≤ 1.998∆2 (Molloy-Reed ‘97)
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Conjecture Conjecture (Erd˝ os-Neˇ setˇril ‘85) For any ¨ graph G with maximum degree ∆, 5 2 ∆ , for even ∆ 0 4 χs (G) ≤ 5 2 1 1 ∆ − 2 ∆ + 4 , for odd ∆ 4 χs0 (G) ≤ 1.998∆2 (Molloy-Reed ‘97) χs0 (G) ≤ 1.93∆2 (Bruhn-Joos ‘15)
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Conjecture Conjecture (Erd˝ os-Neˇ setˇril ‘85) For any ¨ graph G with maximum degree ∆, 5 2 ∆ , for even ∆ 0 4 χs (G) ≤ 5 2 1 1 ∆ − 2 ∆ + 4 , for odd ∆ 4 χs0 (G) ≤ 1.998∆2 (Molloy-Reed ‘97) χs0 (G) ≤ 1.93∆2 (Bruhn-Joos ‘15) χs0 (G) ≤ 1.835∆2 (Bonamy-Perrett-Postle ‘17)
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Conjecture Conjecture (Erd˝ os-Neˇ setˇril ‘85) For any ¨ graph G with maximum degree ∆, 5 2 ∆ , for even ∆ 0 4 χs (G) ≤ 5 2 1 1 ∆ − 2 ∆ + 4 , for odd ∆ 4 χs0 (G) ≤ 1.998∆2 (Molloy-Reed ‘97) χs0 (G) ≤ 1.93∆2 (Bruhn-Joos ‘15) χs0 (G) ≤ 1.835∆2 (Bonamy-Perrett-Postle ‘17) Proven for ∆ = 3 (Andersen ‘92, Horák-Qing-Trotter ‘93)
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Conjecture Conjecture (Erd˝ os-Neˇ setˇril ‘85) For any ¨ graph G with maximum degree ∆, 5 2 ∆ , for even ∆ 0 4 χs (G) ≤ 5 2 1 1 ∆ − 2 ∆ + 4 , for odd ∆ 4 χs0 (G) ≤ 1.998∆2 (Molloy-Reed ‘97) χs0 (G) ≤ 1.93∆2 (Bruhn-Joos ‘15) χs0 (G) ≤ 1.835∆2 (Bonamy-Perrett-Postle ‘17) Proven for ∆ = 3 (Andersen ‘92, Horák-Qing-Trotter ‘93) For ∆ = 4, χs0 (G) ≤ 22 (Cranston ‘06)
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Conjecture Conjecture (Erd˝ os-Neˇ setˇril ‘85) For any ¨ graph G with maximum degree ∆, 5 2 ∆ , for even ∆ 0 4 χs (G) ≤ 5 2 1 1 ∆ − 2 ∆ + 4 , for odd ∆ 4 χs0 (G) ≤ 1.998∆2 (Molloy-Reed ‘97) χs0 (G) ≤ 1.93∆2 (Bruhn-Joos ‘15) χs0 (G) ≤ 1.835∆2 (Bonamy-Perrett-Postle ‘17) Proven for ∆ = 3 (Andersen ‘92, Horák-Qing-Trotter ‘93) For ∆ = 4, χs0 (G) ≤ 22 (Cranston ‘06) For ∆ = 4, χs0 (G) ≤ 21 (Huang-S-Yu ‘17++)
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Proof Sketch Theorem (Huang-S-Yu ‘17++) If G is a multigraph with ∆(G) ≤ 4, then χs0 (G) ≤ 21.
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Proof Sketch Theorem (Huang-S-Yu ‘17++) If G is a multigraph with ∆(G) ≤ 4, then χs0 (G) ≤ 21. Among all counterexamples, choose G so that |V(G)| + |E(G)| is minimized.
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Proof Sketch Theorem (Huang-S-Yu ‘17++) If G is a multigraph with ∆(G) ≤ 4, then χs0 (G) ≤ 21. Among all counterexamples, choose G so that |V(G)| + |E(G)| is minimized. So ∆(G) ≤ 4 and χs0 (G) > 21.
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Proof Sketch Properties of a Minimal Counterexample G G is 4-regular, simple, etc.
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Proof Sketch Properties of a Minimal Counterexample G G is 4-regular, simple, etc. G has girth at least 6.
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Proof Sketch Properties of a Minimal Counterexample G G is 4-regular, simple, etc. G has girth at least 6. G has no edge-cut of size at most 3.
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Proof Sketch Properties of a Minimal Counterexample G G is 4-regular, simple, etc. G has girth at least 6. G has no edge-cut of size at most 3. How to Color the Edges of G
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Proof Sketch Properties of a Minimal Counterexample G G is 4-regular, simple, etc. G has girth at least 6. G has no edge-cut of size at most 3. How to Color the Edges of G Partition the vertices of G into three sets (L, M, and R), where M is a cut-set
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Proof Sketch Properties of a Minimal Counterexample G G is 4-regular, simple, etc. G has girth at least 6. G has no edge-cut of size at most 3. How to Color the Edges of G Partition the vertices of G into three sets (L, M, and R), where M is a cut-set Show that M contains some special vertices.
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Proof Sketch Properties of a Minimal Counterexample G G is 4-regular, simple, etc. G has girth at least 6. G has no edge-cut of size at most 3. How to Color the Edges of G Partition the vertices of G into three sets (L, M, and R), where M is a cut-set Show that M contains some special vertices. Case analysis and color.
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Open Problems Conjecture (Erd˝ os-Neˇ setˇril ‘85) If ∆(G) ≤ 4, then χs0 (G) ≤ 20.
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Open Problems Conjecture (Erd˝ os-Neˇ setˇril ‘85) If ∆(G) ≤ 4, then χs0 (G) ≤ 20. Conjecture (Faudree-Gyárfás-Schelp-Tuza ‘90) Suppose G is a bipartite graph with maximum degree ∆.
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Open Problems Conjecture (Erd˝ os-Neˇ setˇril ‘85) If ∆(G) ≤ 4, then χs0 (G) ≤ 20. Conjecture (Faudree-Gyárfás-Schelp-Tuza ‘90) Suppose G is a bipartite graph with maximum degree ∆. 1
χs0 (G) ≤ ∆2 .
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Open Problems Conjecture (Erd˝ os-Neˇ setˇril ‘85) If ∆(G) ≤ 4, then χs0 (G) ≤ 20. Conjecture (Faudree-Gyárfás-Schelp-Tuza ‘90) Suppose G is a bipartite graph with maximum degree ∆. 1
χs0 (G) ≤ ∆2 .
2
If ∆ ≤ 3 and G has girth at least six, then χs0 (G) ≤ 7
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Open Problems Conjecture (Erd˝ os-Neˇ setˇril ‘85) If ∆(G) ≤ 4, then χs0 (G) ≤ 20. Conjecture (Faudree-Gyárfás-Schelp-Tuza ‘90) Suppose G is a bipartite graph with maximum degree ∆. 1
χs0 (G) ≤ ∆2 .
2
If ∆ ≤ 3 and G has girth at least six, then χs0 (G) ≤ 7
3
If ∆ ≤ 3 and G has ‘large’ girth, then χs0 (G) ≤ 5
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Open Problems Conjecture (Erd˝ os-Neˇ setˇril ‘85) If ∆(G) ≤ 4, then χs0 (G) ≤ 20. Conjecture (Faudree-Gyárfás-Schelp-Tuza ‘90) Suppose G is a bipartite graph with maximum degree ∆. 1
χs0 (G) ≤ ∆2 .
2
If ∆ ≤ 3 and G has girth at least six, then χs0 (G) ≤ 7
3
If ∆ ≤ 3 and G has ‘large’ girth, then χs0 (G) ≤ 5
Theorem (Faudree et al. ‘90) If G is a planar graph with maximum degree ∆, then 4∆ − 4 ≤ χs0 (G) ≤ 4∆ + 4. 13 / 16
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MIGHTY LVIII Grand Valley State University October 6-7, 2017 Plenary Speakers: Doug West David Galvin www.gvsu.edu/math/mighty-lviii MIGHTY_LVIII@ gvsu.edu
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Thanks for your attention!
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Strong chromatic index of graphs with maximum degree four Michael Santana
Joint Work with M. Huang and G. Yu May 2017
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