dedicated to the memory of dear friend Dick Schelp (1936-2010) and - PowerPoint PPT Presentation

dedicated to the memory of dear friend Dick Schelp (1936-2010) and remembering my years in Louisville (1986-1987, 1994-2001) Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 1 / 29

from the irregularity strength of graphs to the degree irregularity of random hypergraphs Jen˝ o Lehel The University of Memphis May 12, 2011 Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 1 / 29

graph: G = ( V , E ) simple: no multiple edges hypergraph: H = ( V , E ) r –uniform hg. | e | = r degree: d ( v ) = |{ e ∈ E | v ∈ e }| (degree) irregular hg. - no degrees repeat d ( x ) � = d ( y ) for any distinct x , y ∈ V there is no irregular simple graph ( 2 -uniform hg.), since a degree must repeat [Behzad, Chartrand, 1967] irregularity strengh of graphs assign pos. integer weights (multiplicities) to the edges such that the weighted degrees are distinct; what is the largest weight we must use? [Chartrand, Jacobson, Lehel, Oellermann, Ruiz, Saba, 1986/1988] Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 2 / 29

Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 3 / 29

0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 1 1 0 1 1 1 1 1 1 r = 3 , n = 6 there exists an irregular simple r-uniform hg. with n vertices, for every r ≥ 3 and n ≥ r + 3 [Gy´ arf´ as, Jacobson, Kinch, Lehel, Schelp, 1989] irregularity of a random hg. what is the probability that a random r -uniform hg has no repeated degrees? Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 4 / 29

content 1 graphs - irregularity strength lower and upper bounds trees regular graphs 2 hypergraphs - degree repetition in random r -uniform hg probability of repeating degrees for r ≥ 6 formula for all r Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 5 / 29

irregularity strength edge k -weighting: w : E − → { 1 , . . . , k } weighted degree: w ( x ) = � { w ( e ) | x ∈ e } irregular weighting: w ( x ) � = w ( y ) for any distinct x , y ∈ V irregularity strength : s ( G ) = min { k | w is an irreg. k -weighting } examples s ( G ) < ∞ provided G has at most 1 isolated vertex and no isolated edge, s ( K 3 ) = 3 , s ( K n ) = 3 , s (2 P 3 ) = 5 , s ( P 4 k ) = 2 k , s ( C 4 k ) = 2 k + 1 Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 6 / 29

Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 7 / 29

lower bound given G and an irregular k -weighting with k = s ( G ), the largest weighted degree is k · ∆( G ) (here ∆( G ) is the max degree in G ) – since all the n weighted degrees are distinct, n ≤ k · ∆( G ), thus we have s ( G ) ≥ n / ∆( G ) – a similarly count gives n i ≤ s ( G ) · i − i + 1, where n i is the number of vertices of degree i in G , and hence s ( G ) ≥ ( n i + i − 1) / i leading to: Proposition. If n i is the number of vertices of degree i in G , then ( n d 0 + n d 0 +1 + · · · + n d ) + d 0 − 1 s ( G ) ≥ max d d 0 ≤ d [Jacobson, Lehel, 1986] Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 8 / 29

upper bounds given G , with n vertices, e edges s ( G ) ≤ 2 e − 1 ”naive” bound ”greedy” bound s ( G ) ≤ 2 n − 3 [Chartrand et al., 1986/1988] ”congruence method” s ( G ) ≤ n + 1 [Aigner, Triesch, 1990] ”spanning tree” bound s ( G ) ≤ n − 1 [Jacobson, Lehel, 1986] for connected G and n ≥ 4 Theorem. If G � = K 3 , and has no isolated edges and vertices, then s ( G ) ≤ n − 1 [Nierhoff, 1998/2000] Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 9 / 29

the congruence method Z m is the additive group (of integers mod m ) reservation on V : a labeling ρ : V → Z m given a forest ( V , F ), a reservation ρ on V is feasible: if there is an edge weighting w : F → { 1 , . . . , m } , call it a realization, such that w ( x ) ≡ ρ ( x ) (mod m ), for every x ∈ V Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 10 / 29

the congruence method given G = ( V , E ) with n vertices, let m = n − 1 , and let ( V , F ) be a spanning forest of G idea: if there was a feasible reservation ρ : V → Z m , then a realization w extended with w ( e ) = m , for every e ∈ E \ F , is an irregular m -weighting, thus s ( G ) ≤ m = n − 1 the proof consists of an algorithm that builds feasible reservations for the subtree components of ( V , F ) this stepwise construction of feasible reservations uses decompositions of Z m because some vertices might have the same reservation, further steps must be done to resolve those repetitions by altering the realization s ( G ) = n − 1 for G = K 4 , 2 P 3 , K 1 , n − 1 , and ... ? Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 11 / 29

bound improvements better bounds if G is regular or G is a tree different from a star Theorem. If G is regular and n ≥ 3, then s ( G ) < n / 2 + 9 [Faudree, Lehel, 1987] Theorem. If a tree T with n ≥ 3 has a matching that contains ℓ leaves, then s ( T ) ≤ n − ℓ , except possibly when ℓ = n / 2 − 1 and T is equipartite. [Aigner, Triesch, 1990] Corollaries. s ( T ) ≤ n − 2 if T is not a star s ( T ) ≤ n − 3 if T is not a star and not a path P n , n ≤ 6, Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 12 / 29

trees if there are n j vertices of degree j , then ( n d 0 + n d 0 +1 + · · · + n d ) + d 0 − 1 s ( G ) ≥ Λ( G ) = max d d 0 ≤ d for a tree T , either Λ( T ) = n 1 or Λ( T ) = ⌈ ( n 1 + n 2 ) / 2 ⌉ . does s ( T ) stay ”close” to Λ( T )? If T is a full d -ary tree, for d = 2 , or 3 then s ( T ) = Λ( T ) = n 1 [Cammack, Schelp, Schrag,1991] If n ≥ 3, and n 2 = 0, then s ( T ) = Λ( T ) = n 1 [Amar, Togni, 1998] If n 1 ≥ 3, and every pair of vertices of degree not equal to 2 are at a distance at least 8, then s ( T ) = Λ( T ) = ⌈ ( n 1 + n 2 ) / 2 ⌉ [Ferrara, Gould, Karo` nski, Pfender, 2010] Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 13 / 29

for several graph families s ( G ) ≤ Λ( G ) + c (paths, cycles, wheels, k -cubes, grids, Tur´ an graphs, union of cliques,...) not true for disconnected graphs: Theorem. Λ( tP 3 ) = 2 t , meanwhile ⌈ (15 t − 1) / 7 ⌉ ≤ s ( tP 3 ) ≤ ⌈ (15 t − 1) / 7 ⌉ + 1 [Kinch,Lehel, 1990] Is s ( T ) ≤ Λ( T ) + c true for connected graphs with some constant c ? In particular, is it true for trees, perhaps with c = 1? [Ebert, Hemmeter, Lazebnik, Woldar, 1990] Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 14 / 29

Theorem. Let T t be obtaned from a path P 2 t +1 by attaching pendant edges to alternating interior vertices. Then Λ( T ) = t + 2 = n 1 , meanwhile √ t →∞ s ( T t ) = 11 − 5 lim · n 1 > 1 . 095 · Λ( T ) 8 [Bohman,Kravitz, 2004] problems find further families of trees with s ( T ) = Λ( T ) find the smallest factor f such that s ( T ) ≤ f · Λ( T ), for all trees Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 15 / 29

regular graphs let G be a d -regular graph with n vertices Proposition. if d = n − 1 then G = K n and s ( G ) = 3 if d ≤ n − 2 then Λ( G ) = ⌈ ( n + d − 1) / d ⌉ ≥ 3 if d = n − 2 then G = K n − n 2 · K 2 and s ( G ) = 3 [Gy´ arf´ as, 1989] Theorem. If d = n − 3 or n − 4, and G � = K 3 , 3 , then s ( G ) = 3 ( s ( K 3 , 3 ) = 4) [Amar, 1988/1993] Theorem. If d = 2 then s ( G ) ≤ ⌈ n / 2 ⌉ + 2 [Faudree, Jacobson,Lehel, Schelp, 1989] Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 16 / 29

problem is there a constant c = c ( d ) such that s ( G ) ≤ n d + c , for all d -regular G ? recall: Λ( G ) = ⌈ n + d − 1 ⌉ [ Faudree, Jacobson, Kinch, Lehel, 1987/1991] d d + 1 , for d ≤ √ n s ( G ) ≤ 48 n d · log n + 1 , for d > √ n s ( G ) ≤ 240 n [Frieze, Gould, Karo` nski, Pfender, 2002] a two-stage strategy was used for the proof 1 first find a ”pre-weighting” w : E → { 1 , 2 , 3 } such that no degree repeats more than M ∼ α · ( n / d ) times (or M ∼ α · ( n / d ) log n -times) /probabilistic stage/ 2 after multiplying each w ( e ) by M there will be enough ”room” to be able to ”shake off” identical degrees by modifying edge weights to obtain an irregular (8 M + 1)-weighting /deterministic Lemma/ Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 17 / 29

similar two-stage strategy with a ”pre-weighting” w : E → { 1 , 2 } leads to d + 6 , for d ≥ 10 4 / 3 · n 2 / 3 · (log n ) 1 / 3 s ( G ) ≤ 48 n [Cuckler, Lazebnik, 2008] the best bound so far is obtained by replacing the probabilistic stage with a deterministic construction: Theorem. If G is d -regular with n vertices, then s ( G ) ≤ 16 n d + 6 [Przybylo, 2009] Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 18 / 29

random uniform hg. almost all hypergraphs have no repeated degrees, i.e. a.a. hg are irregular [Gy´ arf´ as, Jacobson, Kinch, Lehel, Schelp, 1992] random r - uniform hg H r ( n , p ): on an n element (labeled) vertex set V � n � each of the r -sets is taken as an edge r independently and uniformly with probability p . Jen˝ o Lehel (U of M) irregularity 24th cumberland/louisville 19 / 29