▁has ▁been ▁conject ured ▁that ▁no ▁others ▁exist . ▁▁ ▁investig ated ▁the ▁non - in cre asing ▁sequences ▁of ▁numbers ▁ ▁with ▁the ▁property ▁that ▁there ▁exists ▁a ▁proper ▁edge ▁color ing ▁of ▁a ▁given ▁graph ▁ ▁with ▁ ▁edges ▁of ▁the ▁first ▁color , ▁ ▁edges ▁of ▁the ▁second ▁color , ▁etc . ▁They ▁observed ▁that , ▁if ▁a ▁sequence ▁ ▁is ▁feas ible ▁in ▁this ▁sense , ▁and ▁is ▁greater ▁in ▁lex ic ographic ▁order ▁than ▁a ▁sequence ▁ ▁with ▁the ▁same ▁sum , ▁then ▁ ▁is ▁also ▁feas ible . ▁For , ▁if ▁ ▁in ▁lex ic ographic ▁order , ▁then ▁ ▁can ▁be ▁transformed ▁into ▁ ▁by ▁a ▁sequence ▁of ▁steps , ▁each ▁of ▁which ▁reduces ▁one ▁of ▁the ▁numbers ▁ ▁by ▁one ▁unit ▁and ▁increases ▁another ▁later ▁number ▁▁ ▁with ▁ ▁by ▁one ▁unit . ▁In ▁terms ▁of ▁edge ▁color ings , ▁starting ▁from ▁a ▁color ing ▁that ▁real izes ▁, ▁each ▁of ▁these ▁same ▁steps ▁may ▁be ▁performed ▁by ▁sw apping ▁colors ▁ ▁and ▁ ▁on ▁a ▁K em pe ▁chain , ▁a ▁maximal ▁path ▁of ▁edges ▁that ▁alternate ▁between ▁the ▁two ▁colors . ▁In ▁particular , ▁any ▁graph ▁has ▁an ▁equ itable ▁edge ▁color ing , ▁an ▁edge ▁color ing ▁with ▁an ▁optimal ▁number ▁of ▁colors ▁in ▁which ▁every ▁two ▁color ▁classes ▁differ ▁in ▁size ▁by ▁at ▁most ▁one ▁unit . ▁ ▁The ▁De ▁Bru ijn – Er d ős ▁theorem ▁may ▁be ▁used ▁to ▁transfer ▁many ▁edge ▁color ing ▁properties ▁of ▁finite ▁graphs ▁to