New Results on Graph Packing
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1 Graph Packing p.1/18 New Results on Graph Packing Hemanshu Kaul hkaul/. University of Illinois at Urbana-Champaign
2 Graph Packing p.2/18 Introduction Let G 1 = (V 1,E 1 ) and G 2 = (V 2,E 2 ) be graphs of order at most n, with maximum degree 1 and 2, respectively.
3 Graph Packing p.2/18 Introduction Let G 1 = (V 1,E 1 ) and G 2 = (V 2,E 2 ) be graphs of order at most n, with maximum degree 1 and 2, respectively. G 1 and G 2 are said to pack if there exist injective mappings of the vertex sets into [n], V i [n] = {1, 2,...,n}, i = 1, 2, such that the images of the edge sets do not intersect.
4 Graph Packing p.2/18 Introduction Let G 1 = (V 1,E 1 ) and G 2 = (V 2,E 2 ) be graphs of order at most n, with maximum degree 1 and 2, respectively. G 1 and G 2 are said to pack if there exist injective mappings of the vertex sets into [n], V i [n] = {1, 2,...,n}, i = 1, 2, such that the images of the edge sets do not intersect. We may assume V 1 = V 2 = n by adding isolated vertices.
5 Graph Packing p.2/18 Introduction Let G 1 = (V 1,E 1 ) and G 2 = (V 2,E 2 ) be graphs of order at most n, with maximum degree 1 and 2, respectively. G 1 and G 2 are said to pack if there exist injective mappings of the vertex sets into [n], V i [n] = {1, 2,...,n}, i = 1, 2, such that the images of the edge sets do not intersect. there exists a bijection V 1 V 2 such that e E 1 e E 2. G 1 is a subgraph of G 2.
6 Graph Packing p.3/18 Examples and Non-Examples C C 5 5
7 Graph Packing p.3/18 Examples and Non-Examples C 5 C 5 Packing
8 Graph Packing p.3/18 Examples and Non-Examples C 5 C 5 Packing K 4,4 4 K2
9 Graph Packing p.3/18 Examples and Non-Examples C 5 C 5 Packing K 4,4 4 K 2 Packing
10 Graph Packing p.4/18 Examples and Non-Examples K 3,3 3 K 2
11 Graph Packing p.4/18 Examples and Non-Examples K 3,3 3 K 2 No Packing
12 Graph Packing p.4/18 Examples and Non-Examples K 3,3 3 K 2 No Packing 2 K 2 K 3
13 Graph Packing p.4/18 Examples and Non-Examples K 3,3 3 K 2 No Packing 2 K 3 2 K No Packing
14 Graph Packing p.5/18 A Common Generalization Hamiltonian Cycle in graph G : Whether the n-cycle C n packs with G.
15 Graph Packing p.5/18 A Common Generalization Hamiltonian Cycle in graph G : Whether the n-cycle C n packs with G. Existence of a subgraph H in G : Whether H packs with G.
16 Graph Packing p.5/18 A Common Generalization Hamiltonian Cycle in graph G : Whether the n-cycle C n packs with G. Existence of a subgraph H in G : Whether H packs with G. Equitable k-coloring of graph G : Whether G packs with k cliques of order n/k.
17 Graph Packing p.5/18 A Common Generalization Hamiltonian Cycle in graph G : Whether the n-cycle C n packs with G. Existence of a subgraph H in G : Whether H packs with G. Equitable k-coloring of graph G : Whether G packs with k cliques of order n/k. Turan-type problems (forbidden subgraphs).
18 Graph Packing p.5/18 A Common Generalization Hamiltonian Cycle in graph G : Whether the n-cycle C n packs with G. Existence of a subgraph H in G : Whether H packs with G. Equitable k-coloring of graph G : Whether G packs with k cliques of order n/k. Turan-type problems (forbidden subgraphs). Ramsey-type problems.
19 Graph Packing p.5/18 A Common Generalization Hamiltonian Cycle in graph G : Whether the n-cycle C n packs with G. Existence of a subgraph H in G : Whether H packs with G. Equitable k-coloring of graph G : Whether G packs with k cliques of order n/k. Turan-type problems (forbidden subgraphs). Ramsey-type problems. most problems in Extremal Graph Theory.
20 Graph Packing p.6/18 A Distinction In packing problems, each member of a large family of graphs contains each member of another large family of graphs.
21 Graph Packing p.6/18 A Distinction In packing problems, each member of a large family of graphs contains each member of another large family of graphs. In subgraph problems, (usually) at least one of the two graphs is fixed.
22 Graph Packing p.6/18 A Distinction In packing problems, each member of a large family of graphs contains each member of another large family of graphs. In subgraph problems, (usually) at least one of the two graphs is fixed. Erdős-Sos Conjecture : Let G be a graph of order n and T be a tree of size k. If e(g) < 2 1 n(n k) then T and G pack.
23 Graph Packing p.7/18 Sauer and Spencer s Packing Theorem Theorem [Sauer + Spencer, 1978] : If < n, then G 1 and G 2 pack.
24 Graph Packing p.7/18 Sauer and Spencer s Packing Theorem Theorem [Sauer + Spencer, 1978] : If < n, then G 1 and G 2 pack. This is sharp. For n even. G 1 = n 2 K 2, a perfect matching on n vertices. G 2 K n 2 +1, or G 2 = K n 2, n 2 with n 2 odd. Then, = n, and G 1 and G 2 do not pack.
25 Graph Packing p.7/18 Sauer and Spencer s Packing Theorem K 3,3 3 K 2 No Packing G 1 = K n 2, n 2 with n 2 odd G 2 = n 2 K 2 2 K 3 2 K No Packing G 1 = n 2 K 2 G 2 K n 2 +1
26 Graph Packing p.8/18 Extending the Sauer-Spencer Theorem Theorem 1 [Kaul + Kostochka, 2005]: If n, then G 1 and G 2 do not pack if and only if one of G 1 and G 2 is a perfect matching and the other either is K n 2, n 2 with n 2 odd or contains K n This result characterizes the extremal graphs for the Sauer-Spencer Theorem. To appear in Combinatorics, Probability and Computing.
27 Graph Packing p.9/18 Extending the Sauer-Spencer Theorem Theorem 1 [Kaul + Kostochka, 2005]: If n, then G 1 and G 2 do not pack if and only if one of G 1 and G 2 is a perfect matching and the other either is K n 2, n 2 with n 2 odd or contains K n This result can also be thought of as a small step towards the well-known Bollobás-Eldridge conjecture. Bollobás-Eldridge Graph Packing Conjecture : If ( 1 + 1)( 2 + 1) n + 1 then G 1 and G 2 pack.
28 Graph Packing p.10/18 Bollobás-Eldridge Conjecture Bollobás-Eldridge Graph Packing Conjecture [1978] : If ( 1 + 1)( 2 + 1) n + 1 then G 1 and G 2 pack. If true, this conjecture would be sharp, and would be a considerable extension of the Hajnal-Szemerédi theorem on equitable colorings. The conjecture has only been proved when 1 2 [Aigner + Brandt (1993), and Alon + Fischer (1996)], or 1 = 3 and n is huge [Csaba + Shokoufandeh + Szemerédi (2003)].
29 Graph Packing p.11/18 Reformulating the Conjecture Let us consider a refinement of the Bollobás-Eldridge Conjecture. Conjecture : For a fixed 0 ɛ 1. If ( 1 + 1)( 2 + 1) n 2 (1 + ɛ) + 1, then G 1 and G 2 pack. For ɛ = 0, this is essentially the Sauer-Spencer Theorem, while ɛ = 1 gives the Bollobás-Eldridge conjecture. For any ɛ > 0 this would improve the Sauer-Spencer result (in a different way than Theorem 1).
30 Graph Packing p.12/18 Towards the Bollobás-Eldridge Conjecture Theorem 2 [Kaul + Kostochka + Yu, 2005+]: For ɛ = 0.2, and 1, 2 400, If ( 1 + 1)( 2 + 1) n 2 (1 + ɛ) + 1, then G 1 and G 2 pack.
31 Graph Packing p.12/18 Towards the Bollobás-Eldridge Conjecture Theorem 2 [Kaul + Kostochka + Yu, 2005+]: For ɛ = 0.2, and 1, 2 400, If ( 1 + 1)( 2 + 1) n 2 (1 + ɛ) + 1, then G 1 and G 2 pack. In other words, Theorem 2 [Kaul + Kostochka + Yu, 2005+]: For 1, 2 400, If ( 1 + 1)( 2 + 1) (0.6)n + 1, then G 1 and G 2 pack. This is work in progress.
32 Graph Packing p.13/18 Some Proof Ideas for Theorem 1 Theorem 1 [Kaul + Kostochka, 2005]: If n, then G 1 and G 2 do not pack if and only if one of G 1 and G 2 is a perfect matching and the other either is K n 2, n 2 with n 2 odd or contains K n We have to analyze the minimal graphs that do not pack (under the condition n ). (G 1,G 2 ) is a critical pair if G 1 and G 2 do not pack, but for each e 1 E(G 1 ), G 1 e 1 and G 2 pack, and for each e 2 E(G 2 ), G 1 and G 2 e 2 pack.
33 Graph Packing p.14/18 Some Proof Ideas for Theorem 1 Each bijection f : V 1 V 2 generates a (multi)graph G f, with V(G f ) = {(u,f(u)) : u V 1 } (u,f(u)) (u,f(u )) uu E 1 or f(u)f(u ) E 2
34 Graph Packing p.14/18 Some Proof Ideas for Theorem 1 Each bijection f : V 1 V 2 generates a (multi)graph G f, with V(G f ) = {(u,f(u)) : u V 1 } (u,f(u)) (u,f(u )) uu E 1 or f(u)f(u ) E 2 f gives G 1 G 2 G f
35 Graph Packing p.15/18 Some Proof Ideas for Theorem 1 (u 1,u 2 )-switch means replace f by f, with f (u) = f(u), u u 1, u 2 f(u 2 ), u = u 1 f(u 1 ), u = u 2
36 Graph Packing p.15/18 Some Proof Ideas for Theorem 1 (u 1,u 2 )-switch means replace f by f, with f (u) = f(u), u u 1, u 2 f(u 2 ), u = u 1 f(u 1 ), u = u 2 2-neighbors of u 1 2-neighbors of u 2
37 Graph Packing p.15/18 Some Proof Ideas for Theorem 1 (u 1,u 2 )-switch means replace f by f, with f (u) = f(u), u u 1, u 2 f(u 2 ), u = u 1 f(u 1 ), u = u 2 u 1 u 2 (u 1, u 2 ) switch u 2 u 1 G f G f
38 Graph Packing p.16/18 Some Proof Ideas for Theorem 1 An important structure that we utilize in our proof is - (u 1,u 2 ; 1, 2)-link is a path of length two (in G f ) from u 1 to u 2 whose first edge is in E 1 and the second edge is in E 2. For e E 1, an e-packing (quasi-packing) of (G 1,G 2 ) is a bijection f between V 1 and V 2 such that e is the only edge in E 1 that shares its incident vertices with an edge from E 2. Such a packing exists for every edge e in a critical pair.
39 Graph Packing p.17/18 Outline of the Proof of Theorem 1 The main tool Lemma 1 : Let (G 1,G 2 ) be a critical pair and n. Given any e E 1, in a e packing of (G 1,G 2 ) with e = u 1 u 1, the following statements are true. (i) For every u u 1, there exists either a unique (u 1,u; 1, 2) link or a unique (u 1,u; 2, 1) link, (ii) there is no (u 1,u 1 ; 1, 2) link or (u 1,u 1 ; 2, 1) link, (iii) 2deg G1 (u 1 )deg G2 (u 1 ) = n. u u 1 u 1
40 Graph Packing p.17/18 Outline of the Proof of Theorem 1 The main tool Lemma 1 : Let (G 1,G 2 ) be a critical pair and n. Given any e E 1, in a e packing of (G 1,G 2 ) with e = u 1 u 1, the following statements are true. (i) For every u u 1, there exists either a unique (u 1,u; 1, 2) link or a unique (u 1,u; 2, 1) link, (ii) there is no (u 1,u 1 ; 1, 2) link or (u 1,u 1 ; 2, 1) link, (iii) 2deg G1 (u 1 )deg G2 (u 1 ) = n. u u 1 u 1
41 Graph Packing p.17/18 Outline of the Proof of Theorem 1 The main tool Lemma 1 : Let (G 1,G 2 ) be a critical pair and n. Given any e E 1, in a e packing of (G 1,G 2 ) with e = u 1 u 1, the following statements are true. (i) For every u u 1, there exists either a unique (u 1,u; 1, 2) link or a unique (u 1,u; 2, 1) link, (ii) there is no (u 1,u 1 ; 1, 2) link or (u 1,u 1 ; 2, 1) link, (iii) 2deg G1 (u 1 )deg G2 (u 1 ) = n. u u 1 u 1
42 Graph Packing p.18/18 Outline of the Proof of Theorem 1 Lemma 2 : If = n and (G 1,G 2 ) is a critical pair, then every component of G i is either K i, i with i odd, or an isolated vertex, or K i +1, i = 1, 2. Lemma 2 allows us to settle the case of : 1 or 2 = 1. Then, we have to give a packing for all remaining pairs of graphs, to eliminate their possibility.
43 Graph Packing p.18/18 Outline of the Proof of Theorem 1 The following Lemma limits the possible remaining pairs of graphs. Lemma 3 : Let 1, 2 > 1 and = n. If (G 1,G 2 ) is a critical pair and the conflicted edge in a quasi-packing belongs to a component H of G 2 isomorphic to K 2, 2, then every component of G 1 sharing vertices with H is K 1, 1. Now, we completely eliminate such graphs. Lemma 4 : Suppose that 1, 2 3 and odd, and = n. If G 1 consists of 2 copies of K 1, 1 and G 2 consists of 1 copies of K 2, 2, then G 1 and G 2 pack.
44 Graph Packing p.18/18 Outline of the Proof of Theorem 1 Now, lets eliminate the only remaining possibility. Lemma 5 : Let 1, 2 > 1 and = n. If every non-trivial component of G i is K i +1, i = 1, 2, then G 1 and G 2 pack. This would complete the proof of Theorem 1.
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
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