Given　an　instance　of　channel　routing　problem　(CRP)　in　VLSI　design,　the　number　of　tracks　used　in　a　solution　for　the　instance　is　called　the　channel　height　of　the　solution.　The　objective　of　CRP　is　to　minimize　the　channel　heights,　that　is,　to　find　a　solution　with　minimum　channel　height.　In　an　instance　of　CRP,　HCG　and　VCG　denote　the　horizontal　and　vertical　constraint　graphs,　respectively.　Let　GM　be　the　graph　obtained　from　HCG　by　adding　edges　whose　ends　are　connected　by　a　directed　path　in　VCG.　Pal　et　al.　first　gave　lower　bounds　on　the　channel　heights　in　terms　of　the　clique　number　of　GM,　and　presented　algorithms　to　find　such　lower　bounds.　In　this　paper,　we　find　some　interesting　theoretic　properties,　about　the　structure　of　the　cliques　in　GM,　which　can　be　used　to　improve　Pal＇s　algorithms.　So　far,　little　is　known　about　upper　bounds　on　the　channel　heights.　We　find　that　CRP　can　be　translated　into　an　orientation　problem　on　HCG　with　arcs　in　VCG　oriented　and　keeping　directed　acyclic,　and　it　is　also　proved　that　the　channel　height　is　determined　by　the　longest　directed　path　in　the　orientation.　Moreover,we　show　that　a　lemma　on　the　lower　bound　in　[2]　is　incorrect　and　thus　another　lemma　is　given　to　modify　it.　©　2012　Springer-Verlag　GmbH.