Rainbow spanning trees in Abelian groups Bill Kinnersley Department - PowerPoint PPT Presentation

Rainbow spanning trees in Abelian groups Bill Kinnersley Department of Mathematics University of Illinois at Urbana-Champaign wkinner2@illinois.edu Joint work with Robert E. Jamison

Rainbow spanning trees Label the vertices of K n with elements of Z n ; label each edge with the sum of its endpoints. Z 4 : 0 - red 1 - blue 2 - green 3 - purple Which 4-vertex trees appear as rainbow trees?

Rainbow spanning trees Label the vertices of K n with elements of Z n ; label each edge with the sum of its endpoints. Z 4 : 0 - red 1 - blue 2 - green 3 - purple Which 4-vertex trees appear as rainbow trees?

Rainbow spanning trees Label the vertices of K n with elements of Z n ; label each edge with the sum of its endpoints. Z 4 : 0 - red 1 - blue 2 - green 3 - purple Which 4-vertex trees appear as rainbow trees?

Rainbow spanning trees Try again; this time, use labels from Z 2 × Z 2 .

Rainbow spanning trees Try again; this time, use labels from Z 2 × Z 2 . Z 2 × Z 2 : 00 - red 01 - blue 10 - green 11 - purple Which 4-vertex trees appear as rainbow trees?

Rainbow spanning trees Try again; this time, use labels from Z 2 × Z 2 . Z 2 × Z 2 : 00 - red 01 - blue 10 - green 11 - purple Which 4-vertex trees appear as rainbow trees?

Rainbow spanning trees Try again; this time, use labels from Z 2 × Z 2 . Z 2 × Z 2 : 00 - red 01 - blue 10 - green 11 - purple Which 4-vertex trees appear as rainbow trees? K 1 , 3 does; P 4 does not.

Rainbow spanning trees Try again; this time, use labels from Z 2 × Z 2 . Z 2 × Z 2 : 00 - red 01 - blue 10 - green 11 - purple Which 4-vertex trees appear as rainbow trees? K 1 , 3 does; P 4 does not. Given an Abelian group A , let K A denote the corresponding edge-colored complete graph. Which trees appear as rainbow spanning trees in K A ?

Iridescent labeling We say that G is A -iridescent if it embeds as a rainbow subgraph in K A .

Iridescent labeling We say that G is A -iridescent if it embeds as a rainbow subgraph in K A . An embedding of G in K A corresponds to an injective labeling λ : V ( G ) → A . For G to be a rainbow subgraph, all edges must have different sums.

Iridescent labeling We say that G is A -iridescent if it embeds as a rainbow subgraph in K A . An embedding of G in K A corresponds to an injective labeling λ : V ( G ) → A . For G to be a rainbow subgraph, all edges must have different sums. An A -iridescent labeling is a labeling of the vertices of G with elements of A such that ◮ no two vertices have the same label ◮ no two edges have the same sum

Iridescent labeling We say that G is A -iridescent if it embeds as a rainbow subgraph in K A . An embedding of G in K A corresponds to an injective labeling λ : V ( G ) → A . For G to be a rainbow subgraph, all edges must have different sums. An A -iridescent labeling is a labeling of the vertices of G with elements of A such that ◮ no two vertices have the same label ◮ no two edges have the same sum G is A -iridescent if and only if G has an A -iridescent labeling.

Iridescent labeling We say that G is A -iridescent if it embeds as a rainbow subgraph in K A . An embedding of G in K A corresponds to an injective labeling λ : V ( G ) → A . For G to be a rainbow subgraph, all edges must have different sums. An A -iridescent labeling is a labeling of the vertices of G with elements of A such that ◮ no two vertices have the same label ◮ no two edges have the same sum G is A -iridescent if and only if G has an A -iridescent labeling. Prior work: Beals-Gallian-Headley-Jungreis [cycles], Valentin [paths, cycles], Zheng [ A = Z k 2 ]

Iridescent labeling A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum

Iridescent labeling A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum Graceful labeling: Label from Z m , where m = | E ( G ) | . No two vertices have the same label. No two edges have the same absolute difference.

Iridescent labeling A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum Graceful labeling: Label from Z m , where m = | E ( G ) | . No two vertices have the same label. No two edges have the same absolute difference. Harmonious labeling: Label from Z m , where m = | E ( G ) | . No two vertices have the same label. No two edges have the same sum.

Iridescent labeling A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum Graceful labeling: Label from Z m , where m = | E ( G ) | . No two vertices have the same label. No two edges have the same absolute difference. Harmonious labeling: Label from Z m , where m = | E ( G ) | . No two vertices have the same label. No two edges have the same sum. Cordial labeling: Label from Abelian group A . Distribution of labels on vertices is balanced. So is distribution of sums on edges.

Cyclic groups A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum

Cyclic groups A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum Theorem (Hovey) Every n-vertex caterpillar is Z n -iridescent.

Cyclic groups A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum Theorem (Hovey) Every n-vertex caterpillar is Z n -iridescent. Proof (sketch). A = Z 13

Cyclic groups A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum Theorem (Hovey) Every n-vertex caterpillar is Z n -iridescent. Proof (sketch). A = Z 13 0

Cyclic groups A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum Theorem (Hovey) Every n-vertex caterpillar is Z n -iridescent. Proof (sketch). A = Z 13 0 1

Cyclic groups A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum Theorem (Hovey) Every n-vertex caterpillar is Z n -iridescent. Proof (sketch). 2 A = Z 13 0 1

Cyclic groups A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum Theorem (Hovey) Every n-vertex caterpillar is Z n -iridescent. Proof (sketch). 2 A = Z 13 0 1 3

Cyclic groups A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum Theorem (Hovey) Every n-vertex caterpillar is Z n -iridescent. Proof (sketch). 2 4 A = Z 13 0 1 3

Cyclic groups A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum Theorem (Hovey) Every n-vertex caterpillar is Z n -iridescent. Proof (sketch). 2 4 A = Z 13 0 1 3 5

Cyclic groups A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum Theorem (Hovey) Every n-vertex caterpillar is Z n -iridescent. Proof (sketch). 2 4 A = Z 13 0 1 3 5 6

Cyclic groups A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum Theorem (Hovey) Every n-vertex caterpillar is Z n -iridescent. Proof (sketch). 7 2 4 A = Z 13 0 1 3 5 6

Cyclic groups A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum Theorem (Hovey) Every n-vertex caterpillar is Z n -iridescent. Proof (sketch). 7 2 4 A = Z 13 0 1 8 3 5 6

Cyclic groups A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum Theorem (Hovey) Every n-vertex caterpillar is Z n -iridescent. Proof (sketch). 7 2 4 A = Z 13 0 1 8 9 3 5 6

Cyclic groups A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum Theorem (Hovey) Every n-vertex caterpillar is Z n -iridescent. Proof (sketch). 7 2 4 A = Z 13 0 1 8 9 10 3 5 6

Cyclic groups A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum Theorem (Hovey) Every n-vertex caterpillar is Z n -iridescent. Proof (sketch). 7 2 11 4 A = Z 13 0 1 8 9 10 3 5 6

Cyclic groups A -iridescent labeling: label vertices of G with elements of A so that ◮ no two vertices have the same label ◮ no two edges have the same sum Theorem (Hovey) Every n-vertex caterpillar is Z n -iridescent. Proof (sketch). 7 2 11 4 12 A = Z 13 0 1 8 9 10 3 5 6