t m Tatra Mt Math Publ Mathematical Publications THE SUBCHROMATIC INDEX OF GRAPHS Jir Fiala Van Bang Le ABSTRACT In an edge coloring of a graph each color class forms a subgraph without a path of length twoamatching An edge sub coloring extends this concept each color class in an edge sub coloring forms a subgraph without path of length three While every graph with maximum degree at most twoisedge sub colorable wepoint out in this pap er that recognizing edge sub colorable graphs with maximum degree three is NPcomplete even when restricted to trianglefree graphs As bypro ducts we obtain NP completeness results for the star index and the sub chromatic numb er for several classes of graphs It is also proved that recognizing edge sub colorable graphs is NP complete Moreover edge sub colorings and sub chromatic index of various basic graph classes are studied In particular we showthatevery unicyclic graph is edge sub colorable and edge sub colorable unicyclic graphs have a simple structure allowing an easy linear time recognition Intro duction Let G V E b e a graph An independent set a clique is a set of pairwise nonadjacent adjacent vertices For W V the subgraph of G induced by W is denoted by GW For F E the symbol V F denotes the set of endvertices of edges from F and GF V F F is the subgraph of G induced by the edge set F A prop er vertex r coloring of G is a partition V V into disjoint in r dep endent sets called color classes of the coloring The chromatic number G is the smallest number r for which G admits a vertex r coloring One of the most interesting generalizations of the classical vertex coloring is the notion of vertex sub coloring see A vertex r subcoloring is a partition V V of V where each color class V consists of disjoint cliques of various r i sizes The smallest number r for which G has a vertex r sub coloring is called the subchromatic number GofG s M a thematics Subject Classification C C Keywordscomputationalcomplexity line graphs sub coloring sub chromatic index Supp orted by the Ministry of Education of the Czech Republic as pro ject M JIRI FIALA VAN BANG LE Note that a partition V V of V isavertex r coloring of G V Eif r and only if for each i the graph GV do es not contain a P as an induced i subgraph The partition is a vertex r sub coloring if and only if for each i the graph GV do es not contain a P as an induced subgraph P denotes the i k path on k vertices A prop er edge r coloring is a partition E E of E into color classes r E in whichevery two distinct edges do not have an endvertex in common ie i each E forms a matching The chromatic index G is the smallest number i r for which G admits an edge r coloring Clearly a partition E E of E r is an edge r coloring of G V E if and only if for each i the subgraph GE i do es not contain a P as a not necessarily induced subgraph This observation leads to the following natural generalization of the classical edge coloring Definition An edge r subcoloring of the edges of a graph G V Eis a partition E E of E into disjoint color classes E such that for each E r i i the graph GE contains no P as a not necessarily induced subgraph The i subchromatic index G is the smallest number r for which G admits an edge s r sub coloring Remark Obviously a partition E E of E G is an edge r sub coloring r of G if and only if for each i the connected comp onents of GE are stars or i triangles where a star is a complete bipartite graph K for some s s A related notion that has b een studied in the literature is as follows A partition E E of E Gisastar partition of G if for each i the connected r comp onents of GE are stars The star index GofG is the smallest i number r for which G has a star partition into r subsets E cf i Clearly G G for all graphs G and it holds that G G s s whenever G is trianglefree Recall that the line graph LG of a graph G has the edges of G as vertices and two distinct edges e e are adjacentin LG whenever they have an endver tex in common It is wellknown that prop er edge colorings of G corresp ond to prop er vertex colorings of LG and vice versa In particular G LG Likewise the following fact is easy to see Fact Edge sub colorings of a graph G corresp ond to vertex sub colorings of the line graph LG of G and vice versa In particular G LG s s Our terminology of edge sub coloring is intended to recall this fact THE SUBCHROMATIC INDEX OF GRAPHS Basic prop erties and examples The sub chromatic index of a disconnected graph is the maximum sub chro matic index among those of its connected comp onents Hence without loss of generalitywe assume throughout this pap er that all graphs are connected By the denition the edge sub chromatic index is monotone with resp ect to graph inclusion ie G G G G The next observation shows a s s close link b etween sub chromatic indices of graphs where one is formed from the other by removal of a vertex Observation For any graph G and anyvertex v of G it holds that G G n v s s Proo f Any sub coloring of G n v can b e extended to a sub coloring of G by using an extra new color on all edges incidentwith v General lower and upp er b ounds for the sub chromatic index are given b elow Let G b e the maximum degree of a vertex in the graph G Proposition Any graph G with m edges on n vertices satises m G G s n m Moreover G if G is trianglefree s n P r o o f Since every color class consists of stars and triangles it may contain at most n edges In the sub coloring each edge has to b e colored and the lower b ound follows Note that a color class in a graph on n vertices can have n edges if and only if n isamultiple of and the class itself is a covering of the vertices by n disjoint triangles In trianglefree graphs no such class exists and the lower b ound can b e made sharp How to obtain the upp er b ound wehavefromFact LG G s s l m l m LG G G where the inequality for the sub chro matic number was shown in To nd a valid sub coloring using at most G edge colors ecientlywe may pro ceed greedily on the vertex set with each new vertex u assign colors to its adjacent edges as follows for an edge u v pick a color that is not used on an already colored edge incident with v Such a sub coloring is trianglefree and all stars have the prop erty that the center of the star is the latest vertex of the star in the order JIRI FIALA VAN BANG LE C Observe that the upp er b ound is attained eg for the cycle s C or the Petersen graph P P The last prop erty follows s for any cubic graph whichcontains C as an induced subgraph it is imp ossible to extend a valid sub coloring to all edges incident to the cycle C See also Figure r G r Moreover Corollary Any r regular graph G satises s r G if G is trianglefree s Trees and cycles For trees and cycles the sub chromatic index can b e determined explicitly as follows Proposition i For anytree T T T T if and only if T is not s s a star ii C and C C for all n s n n s P r o o f Color greedily Cacti A cactus is a connected graph in whichevery blo ck maximal connected sub graph is an edge or a cycle Equivalently a graph G is a cactus if and only if every two cycles in G are edgedisjoint Proposition For all cacti G wehave G G Moreover an s edge sub coloring can b e found in linear time Proo f Let T b e a breadthrst search bfs tree of G ro oted at vertex v We claim that all edges of G outside T form a matching During the searchwe arrange vertices into levels based on the distance from the initial vertex Since G is a cactus the tree T misses from each o dd cycle the edge connecting the twovertices at the highest level Similarly for an even cycle one of the two edges incident with the vertex at the highest level remains outside T Now observe the given twoedges e e of E G n E T either e is separated from e bythelowest vertex of the cycle containing e or viceversa We color T with two colors and use the matching E G n E T as the third color This shows G Since a bfstree can b e p erformed in linear time Prop osition follows THE SUBCHROMATIC INDEX OF GRAPHS Weleave it as an op en problem whether cacti with sub chromatic index at most allow a simple structural description Since all cacti have treewidth up p erb ounded bytheirsubchromatic index can b e computed in p olynomial time as we will show later in Section Observe that in any edge sub colored cactus G eachvertex of degree at least is either a center of a mono chromatic star or it b elongs to a mono chro matic triangle Let us direct the edges of mono chromatic stars K k k towards their centers The other edges remain undirected Clearlynovertex of degree at least in G is of outdegree or more Each directed cycle is of even length since the colors of stars must alternate Figure shows some cacti G that do allowsuch orientation of their cycles G and hence have s C k k Figure Some cacti with sub chromatic index Unicycli c graphs A connected graph is unicyclic if it contains exactly one cycle As we show now all edge sub colorable unicyclic graphs have a simple structure and hence can b e easily recognized in linear time Theorem For any unicyclic graph G wehave G Moreover s G if and only if the only cycle C of G has length k k s and all vertices of C are of degree at least in G P r o o f Since unicyclic graphs are cacti the rst part follows from Prop o sition Wehave shown ab ove that if an o dd cycle C k contains no k vertex of degree