connected , ▁and ▁if ▁deleting ▁any ▁mon och rom atic ▁pair ▁of ▁edges ▁leads ▁to ▁a ▁sub graph ▁that ▁is ▁still ▁non - b ipart ite . ▁These ▁conditions ▁may ▁all ▁be ▁tested ▁easily ▁in ▁polynomial ▁time ; ▁however , ▁the ▁problem ▁of ▁testing ▁whether ▁a ▁ 4 - edge - color ed ▁ 4 - regular ▁graph ▁has ▁a ▁drawing ▁with ▁edges ▁of ▁four ▁s lop es , ▁representing ▁the ▁colors ▁by ▁s lop es , ▁is ▁complete ▁for ▁the ▁exist ential ▁theory ▁of ▁the ▁re als , ▁a ▁complexity ▁class ▁at ▁least ▁as ▁difficult ▁as ▁being ▁N P - complete . ▁ ▁As ▁well ▁as ▁being ▁related ▁to ▁the ▁maximum ▁degree ▁and ▁maximum ▁matching ▁number ▁of ▁a ▁graph , ▁the ▁chrom atic ▁index ▁is ▁closely ▁related ▁to ▁the ▁linear ▁ar bor icity ▁ ▁of ▁a ▁graph ▁, ▁the ▁minimum ▁number ▁of ▁linear ▁for ests ▁( dis joint ▁un ions ▁of ▁paths ) ▁into ▁which ▁the ▁graph ' s ▁edges ▁may ▁be ▁partition ed . ▁A ▁matching ▁is ▁a ▁special ▁kind ▁of ▁linear ▁forest , ▁and ▁in ▁the ▁other ▁direction , ▁any ▁linear ▁forest ▁can ▁be ▁ 2 - edge - color ed , ▁so ▁for ▁every ▁ ▁it ▁follows ▁that ▁. ▁A ki y ama ' s ▁conject ure ▁( named ▁for ▁Jin ▁A ki y ama ) ▁states ▁that ▁, ▁from ▁which ▁it ▁would ▁follow ▁more ▁strongly ▁that ▁. ▁For ▁graphs ▁of ▁maximum ▁degree ▁three , ▁ ▁is ▁always ▁exactly ▁two , ▁so ▁in ▁this ▁case ▁the ▁bound ▁ ▁matches