rainboset E R E V E T with an injection R V ee R Tle 7 e sa m P - PDF document

RMM 2020 Day 1 Riddle Problem 3 and other short stories Rainbow odd cycles Ron Holzman Ron Aharoni with joint work Joseph Briggs multi sett E of sets an E Given a family DEI rainboset E R E V E T with an injection R V ee R Tle 7 e sa m P a property P find smallest Given Problee m For every family 8 if the following holds ar m and every member of E satisfies 70 then I El a E rainbow set R with P 7 Bairam 821 Colorful Caratheldory theorem Example subsets of R Every family of each containing a

convex hull in its has a rainbow set with the same property I Drisko 98 Aharoni Berger is in in any bipartite graph i matchings of size 2n a rainbow watching of site have n subsets of IR each 7 Is in its cow hull Remark n n do NOT have rainbow set with the same a generically property 2 matchings of size Aharoni Kotar Ziv 18 Zn n in any bipartite graph Dd have a rainbow matching of UNLESS size a YI edge disjoint subwaPohsdd cycles www.T stable cycles of ccondo matchings in bipartite graph Every family of n odd cycles in Ku has a rainbow Riddle odd cycle show sharpness of IT IT Remake n i 4 is odd works only when h

A family 0 of cycles is a primed EH A cactus if all the cycles are V 1 HI 8 a cycle on 101 1 vertices identical to 01 O can be partitioned into 2 If UI UI Or sur pruned cacti 0 shave exactly one vertex odd cycles has If a family in of no Tim n k n 11 a pruned cactus it is rainbow odd cycle then n i odd cycles every family of h is 001 When even has a rainbow in Ku cannot be a pruned calctus odd cycle El Oltl Suppose I VWOH ProofsketchofTHM i Break into 3 eases SH IV UK I E I KI t l Cased There exists K O Every odd cycle in 0 is Hamiltonian Casey IV UK I KI t l For every K 0 Case some Oi C O is not Hamiltonian and Say On is not Ham On O Oz Oi proofofcase3 O Vl Knoll 9h V T on Oh if Oulu Consider 0 v

f K E Q1 connected subgraphs ar are They IV UK'll 3 I k't t l them 01 has R spanning V rain a Using On rainbow odd cycle Ba 7 Rainbow cycles n cycles in kn has a rainbow cycle Every family of Prof Remark i Pruned cacti give sharpness T recursively For every family 0 of nicycles in kn no rainbow The cycle exists iff O a saguaro is Edge disjoint subgraphs non empty Prof Every family of n edge disjoint leaf link a rainbow cycle subgraphs in kn has GI u E E E u Remake G

Therefor Every family of n i edge disjoint nonempty subgraphs the family is no rainbow cycles exists in kn link leaf it Even cycles cannot improve beyond E n seven cycles in Ku Every family of L3h J Prof a rainbow even cycle has