SLIDE 1
Definitions and Examples
- Def. injective coloring: vertex coloring such that if u and v have a
Injective coloring of sparse graphs Daniel W. Cranston DIMACS, - - PowerPoint PPT Presentation
Injective coloring of sparse graphs Daniel W. Cranston DIMACS, - - PowerPoint PPT Presentation
Injective coloring of sparse graphs Daniel W. Cranston DIMACS, Rutgers and Bell Labs joint with Seog-Jin Kim and Gexin Yu dcransto@dimacs.rutgers.edu AMS Meeting, University of Illinois March 28, 2009 Definitions and Examples Def. injective
SLIDE 2
SLIDE 3
Definitions and Examples
- Def. injective coloring: vertex coloring such that if u and v have a
common neighbor, then c(u) = c(v). injective chromatic number: χi(G) is minimum k such that G has an injective coloring with k colors.
SLIDE 4
Definitions and Examples
- Def. injective coloring: vertex coloring such that if u and v have a
common neighbor, then c(u) = c(v). injective chromatic number: χi(G) is minimum k such that G has an injective coloring with k colors.
SLIDE 5
Definitions and Examples
- Def. injective coloring: vertex coloring such that if u and v have a
common neighbor, then c(u) = c(v). injective chromatic number: χi(G) is minimum k such that G has an injective coloring with k colors. Easy bounds: ∆ ≤ χi(G) ≤ ∆2 − ∆ + 1
SLIDE 6
Definitions and Examples
- Def. injective coloring: vertex coloring such that if u and v have a
common neighbor, then c(u) = c(v). injective chromatic number: χi(G) is minimum k such that G has an injective coloring with k colors. Easy bounds: ∆ ≤ χi(G) ≤ ∆2 − ∆ + 1 (greedy)
SLIDE 7
Definitions and Examples
- Def. injective coloring: vertex coloring such that if u and v have a
common neighbor, then c(u) = c(v). injective chromatic number: χi(G) is minimum k such that G has an injective coloring with k colors. Easy bounds: ∆ ≤ χi(G) ≤ ∆2 − ∆ + 1 (greedy)
SLIDE 8
Definitions and Examples
- Def. injective coloring: vertex coloring such that if u and v have a
common neighbor, then c(u) = c(v). injective chromatic number: χi(G) is minimum k such that G has an injective coloring with k colors. Easy bounds: ∆ ≤ χi(G) ≤ ∆2 − ∆ + 1 (greedy)
SLIDE 9
Sparse graphs and Mad(G)
Ques. When can we prove χi(G) ≤ ∆ + c for c ∈ {0, 1, 2}?
SLIDE 10
Sparse graphs and Mad(G)
Ques. When can we prove χi(G) ≤ ∆ + c for c ∈ {0, 1, 2}? We will study sparse graphs.
SLIDE 11
Sparse graphs and Mad(G)
Ques. When can we prove χi(G) ≤ ∆ + c for c ∈ {0, 1, 2}? We will study sparse graphs.
- Def. maximum average degree of G Mad(G) = maxH⊆G
2E(H) V (H)
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Sparse graphs and Mad(G)
Ques. When can we prove χi(G) ≤ ∆ + c for c ∈ {0, 1, 2}? We will study sparse graphs.
- Def. maximum average degree of G Mad(G) = maxH⊆G
2E(H) V (H)
Ex. planar graphs Mad(G) < 6
SLIDE 13
Sparse graphs and Mad(G)
Ques. When can we prove χi(G) ≤ ∆ + c for c ∈ {0, 1, 2}? We will study sparse graphs.
- Def. maximum average degree of G Mad(G) = maxH⊆G
2E(H) V (H)
Ex. planar graphs Mad(G) < 6 forests Mad(G) < 2
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Sparse graphs and Mad(G)
Ques. When can we prove χi(G) ≤ ∆ + c for c ∈ {0, 1, 2}? We will study sparse graphs.
- Def. maximum average degree of G Mad(G) = maxH⊆G
2E(H) V (H)
Ex. planar graphs Mad(G) < 6 forests Mad(G) < 2 planar, girth ≥ g Mad(G) <
2g g−2
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∆ = 3 and Mad(G) < 36
13
Thm. If ∆ = 3 and Mad(G) < 36
13, then χi(G) ≤ 5.
SLIDE 16
∆ = 3 and Mad(G) < 36
13
Thm. If ∆ = 3 and Mad(G) < 36
13, then χi(G) ≤ 5.
Pf. Assume G is min counterexample. Forbidden configs.
SLIDE 17
∆ = 3 and Mad(G) < 36
13
Thm. If ∆ = 3 and Mad(G) < 36
13, then χi(G) ≤ 5.
Pf. Assume G is min counterexample. Forbidden configs.
SLIDE 18
∆ = 3 and Mad(G) < 36
13
Thm. If ∆ = 3 and Mad(G) < 36
13, then χi(G) ≤ 5.
Pf. Assume G is min counterexample. Forbidden configs.
SLIDE 19
∆ = 3 and Mad(G) < 36
13
Thm. If ∆ = 3 and Mad(G) < 36
13, then χi(G) ≤ 5.
Pf. Assume G is min counterexample. Forbidden configs.
SLIDE 20
∆ = 3 and Mad(G) < 36
13
Thm. If ∆ = 3 and Mad(G) < 36
13, then χi(G) ≤ 5.
Pf. Assume G is min counterexample. Forbidden configs.
SLIDE 21
∆ = 3 and Mad(G) < 36
13
Thm. If ∆ = 3 and Mad(G) < 36
13, then χi(G) ≤ 5.
Pf. Assume G is min counterexample. Forbidden configs. 2 1 1 2
SLIDE 22
∆ = 3 and Mad(G) < 36
13
Thm. If ∆ = 3 and Mad(G) < 36
13, then χi(G) ≤ 5.
Pf. Assume G is min counterexample. Forbidden configs.
SLIDE 23
∆ = 3 and Mad(G) < 36
13
Thm. If ∆ = 3 and Mad(G) < 36
13, then χi(G) ≤ 5.
Pf. Assume G is min counterexample. Forbidden configs. Discharging rules R1) Each 3-vertex gives
3 13 to each 2-vertex at distance 1.
R2) Each 3-vertex gives
1 13 to each 2-vertex at distance 2.
SLIDE 24
∆ = 3 and Mad(G) < 36
13
Thm. If ∆ = 3 and Mad(G) < 36
13, then χi(G) ≤ 5.
Pf. Assume G is min counterexample. Forbidden configs. Discharging rules R1) Each 3-vertex gives
3 13 to each 2-vertex at distance 1.
R2) Each 3-vertex gives
1 13 to each 2-vertex at distance 2.
µ(v) = d(v) and we check that µ∗(v) ≥ 36
13 for all v.
SLIDE 25
∆ = 3 and Mad(G) < 36
13
Thm. If ∆ = 3 and Mad(G) < 36
13, then χi(G) ≤ 5.
Pf. Assume G is min counterexample. Forbidden configs. Discharging rules R1) Each 3-vertex gives
3 13 to each 2-vertex at distance 1.
R2) Each 3-vertex gives
1 13 to each 2-vertex at distance 2.
µ(v) = d(v) and we check that µ∗(v) ≥ 36
13 for all v.
3-vertex not adjacent to 2-vertex: 3-vertex adjacent to 2-vertex: 2-vertex:
SLIDE 26
∆ = 3 and Mad(G) < 36
13
Thm. If ∆ = 3 and Mad(G) < 36
13, then χi(G) ≤ 5.
Pf. Assume G is min counterexample. Forbidden configs. Discharging rules R1) Each 3-vertex gives
3 13 to each 2-vertex at distance 1.
R2) Each 3-vertex gives
1 13 to each 2-vertex at distance 2.
µ(v) = d(v) and we check that µ∗(v) ≥ 36
13 for all v.
3-vertex not adjacent to 2-vertex: µ∗(v) ≥ 3 − 3( 1
13) = 36 13
3-vertex adjacent to 2-vertex: 2-vertex:
SLIDE 27
∆ = 3 and Mad(G) < 36
13
Thm. If ∆ = 3 and Mad(G) < 36
13, then χi(G) ≤ 5.
Pf. Assume G is min counterexample. Forbidden configs. Discharging rules R1) Each 3-vertex gives
3 13 to each 2-vertex at distance 1.
R2) Each 3-vertex gives
1 13 to each 2-vertex at distance 2.
µ(v) = d(v) and we check that µ∗(v) ≥ 36
13 for all v.
3-vertex not adjacent to 2-vertex: µ∗(v) ≥ 3 − 3( 1
13) = 36 13
3-vertex adjacent to 2-vertex: µ∗(v) = 3 − 3
13 = 36 13
2-vertex:
SLIDE 28
∆ = 3 and Mad(G) < 36
13
Thm. If ∆ = 3 and Mad(G) < 36
13, then χi(G) ≤ 5.
Pf. Assume G is min counterexample. Forbidden configs. Discharging rules R1) Each 3-vertex gives
3 13 to each 2-vertex at distance 1.
R2) Each 3-vertex gives
1 13 to each 2-vertex at distance 2.
µ(v) = d(v) and we check that µ∗(v) ≥ 36
13 for all v.
3-vertex not adjacent to 2-vertex: µ∗(v) ≥ 3 − 3( 1
13) = 36 13
3-vertex adjacent to 2-vertex: µ∗(v) = 3 − 3
13 = 36 13
2-vertex: 2 + 2( 3
13) + 4( 1 13) = 36 13
SLIDE 29
∆ = 3 and Mad(G) < 36
13
Thm. If ∆ = 3 and Mad(G) < 36
13, then χi(G) ≤ 5.
Pf. Assume G is min counterexample. Forbidden configs. Discharging rules R1) Each 3-vertex gives
3 13 to each 2-vertex at distance 1.
R2) Each 3-vertex gives
1 13 to each 2-vertex at distance 2.
µ(v) = d(v) and we check that µ∗(v) ≥ 36
13 for all v.
3-vertex not adjacent to 2-vertex: µ∗(v) ≥ 3 − 3( 1
13) = 36 13
3-vertex adjacent to 2-vertex: µ∗(v) = 3 − 3
13 = 36 13
2-vertex: 2 + 2( 3
13) + 4( 1 13) = 36 13
And the theorem is best possible.
SLIDE 30
∆ = 3 and Mad(G) < 36
13
Thm. If ∆ = 3 and Mad(G) < 36
13, then χi(G) ≤ 5.
Pf. Assume G is min counterexample. Forbidden configs. Discharging rules R1) Each 3-vertex gives
3 13 to each 2-vertex at distance 1.
R2) Each 3-vertex gives
1 13 to each 2-vertex at distance 2.
µ(v) = d(v) and we check that µ∗(v) ≥ 36
13 for all v.
3-vertex not adjacent to 2-vertex: µ∗(v) ≥ 3 − 3( 1
13) = 36 13
3-vertex adjacent to 2-vertex: µ∗(v) = 3 − 3
13 = 36 13
2-vertex: 2 + 2( 3
13) + 4( 1 13) = 36 13
And the theorem is best possible.
SLIDE 31
∆ = 3 and Mad(G) < 36
13
Thm. If ∆ = 3 and Mad(G) < 36
13, then χi(G) ≤ 5.
Pf. Assume G is min counterexample. Forbidden configs. Discharging rules R1) Each 3-vertex gives
3 13 to each 2-vertex at distance 1.
R2) Each 3-vertex gives
1 13 to each 2-vertex at distance 2.
µ(v) = d(v) and we check that µ∗(v) ≥ 36
13 for all v.
3-vertex not adjacent to 2-vertex: µ∗(v) ≥ 3 − 3( 1
13) = 36 13
3-vertex adjacent to 2-vertex: µ∗(v) = 3 − 3
13 = 36 13
2-vertex: 2 + 2( 3
13) + 4( 1 13) = 36 13
And the theorem is best possible.
SLIDE 32
∆ = 3 and Mad(G) < 5
2
Thm. If ∆ = 3 and Mad(G) < 5
2, then χi(G) ≤ 4.
SLIDE 33
∆ = 3 and Mad(G) < 5
2
Thm. If ∆ = 3 and Mad(G) < 5
2, then χi(G) ≤ 4.
Pf. Assume G is min counterexample. Forbidden configs.
SLIDE 34
∆ = 3 and Mad(G) < 5
2
Thm. If ∆ = 3 and Mad(G) < 5
2, then χi(G) ≤ 4.
Pf. Assume G is min counterexample. Forbidden configs.
SLIDE 35
∆ = 3 and Mad(G) < 5
2
Thm. If ∆ = 3 and Mad(G) < 5
2, then χi(G) ≤ 4.
Pf. Assume G is min counterexample. Forbidden configs. Discharging: µ(v) = d(v).
SLIDE 36
∆ = 3 and Mad(G) < 5
2
Thm. If ∆ = 3 and Mad(G) < 5
2, then χi(G) ≤ 4.
Pf. Assume G is min counterexample. Forbidden configs. Discharging: µ(v) = d(v). We want G to have no more 2-vertices than 3-vertices. Why?
SLIDE 37
∆ = 3 and Mad(G) < 5
2
Thm. If ∆ = 3 and Mad(G) < 5
2, then χi(G) ≤ 4.
Pf. Assume G is min counterexample. Forbidden configs. Discharging: µ(v) = d(v). We want G to have no more 2-vertices than 3-vertices. Why? Consider G2, subgraph of edges incident to 2-vertices.
SLIDE 38
∆ = 3 and Mad(G) < 5
2
Thm. If ∆ = 3 and Mad(G) < 5
2, then χi(G) ≤ 4.
Pf. Assume G is min counterexample. Forbidden configs. Discharging: µ(v) = d(v). We want G to have no more 2-vertices than 3-vertices. Why? Consider G2, subgraph of edges incident to 2-vertices. We win if each component of G2 is a tree or cycle.
SLIDE 39
∆ = 3 and Mad(G) < 5
2
Thm. If ∆ = 3 and Mad(G) < 5
2, then χi(G) ≤ 4.
Pf. Assume G is min counterexample. Forbidden configs. Discharging: µ(v) = d(v). We want G to have no more 2-vertices than 3-vertices. Why? Consider G2, subgraph of edges incident to 2-vertices. We win if each component of G2 is a tree or cycle.
SLIDE 40
∆ = 3 and Mad(G) < 5
2 (again)
SLIDE 41
∆ = 3 and Mad(G) < 5
2 (again)
SLIDE 42
∆ = 3 and Mad(G) < 5
2 (again)
Lemma (Vizing) For a connected graph G, let L be a list assignment such that |L(v)| ≥ d(v) for all v. G is L-colorable if a) |L(y)| > d(y) for some vertex y; or b) G is 2-connected and the lists are not all identical.
SLIDE 43
∆ = 3 and Mad(G) < 5
2 (again)
Lemma (Vizing) For a connected graph G, let L be a list assignment such that |L(v)| ≥ d(v) for all v. G is L-colorable if a) |L(y)| > d(y) for some vertex y; or b) G is 2-connected and the lists are not all identical.
SLIDE 44
∆ = 3 and Mad(G) < 5
2 (again)
Lemma (Vizing) For a connected graph G, let L be a list assignment such that |L(v)| ≥ d(v) for all v. G is L-colorable if a) |L(y)| > d(y) for some vertex y; or b) G is 2-connected and the lists are not all identical.
SLIDE 45
∆ = 3 and Mad(G) < 5
2 (again)
Lemma (Vizing) For a connected graph G, let L be a list assignment such that |L(v)| ≥ d(v) for all v. G is L-colorable if a) |L(y)| > d(y) for some vertex y; or b) G is 2-connected and the lists are not all identical.
SLIDE 46