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On the Edge-Colouring of Split Graphs

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On the Edge-Colouring of Split Graphs O conteudo do presente relatorio e de unica resp onsabilidade dos autores The contents of this rep ort are the sole resp onsibility of the authors On the edgecolouring of split graphs Celina M H de Figueiredo Joao Meidanis y Celia Picinin de Mel lo Relatorio Tecnico IC Junho de On the edgecolouring of split graphs y y Celina M H de Figueiredo Joao Meidanis Celia Picinin de Mello Abstract We consider the following question can split graphs with o dd maximum degree b e edgecoloured with maximum degree colours We show that any o dd maxi mum degree split graph can b e transformed into a sp ecial split graph For this sp ecial split graph we were able to solve the question in case the graph has a quasiuniversal vertex Sumario Consideramos o seguinte problema grafos split com grau maximo mpar admitem uma coloracao de arestas com grau maximo cores Mostramos como qualquer grafo split com grau maximo mpar p o de ser transformado em um grafo split esp ecial Para este grafo split esp ecial resolvemos o problema prop osto no caso do grafo admitir um vertice quaseuniversal Intro duction An edgecolouring of a graph is an assignment of colours to its edges such that no adjacent edges have the same colour The chromatic index of a graph is the minimum numb er of colours required to pro duce an edgecolouring for that graph An easy lower b ound for the chromatic index is the maximum vertex degree A celebrated theorem by Vizing states that these two quantities dier by at most one Graphs whose chromatic index equals the maximum degree are said to b e Class graphs whose chromatic index exceeds the maximum degree by one are said to b e Class Very little is known ab out the complexity of computing the chromatic index in general It is known that complete graphs and chordless cycles are Class if and only if the numb er of vertices is even Bipartite graphs are all Class Planthold proved that graphs with a universal vertex or with a quasiuniversal vertex can b e classied in p olynomial time Homan and Ro dger derived a similar result for complete multipartite graphs The fourcolour problem is equivalent to saying that all connected regular planar graphs are Class In previous pap ers we settled the question for doubly chordal graphs with o dd maximum degree and for indierence graphs with at most three maximal cliques Doubly chordal graphs include interval graphs and strongly chordal graphs These results summarize all that is known ab out edgecolouring sp ecic classes of graphs In this note we consider the following question can split graphs with o dd maximum degree b e edgecoloured with colours Edgecolouring of split graphs was rst considered Instituto de Matematica Universidade Federal do Rio de Janeiro Caixa Postal Rio de Janeiro RJ Brasil celinacosufrjbr Partially supp orted by CNPq y Instituto de Computacao Universidade Estadual de Campinas Caixa Postal Campinas SP Brasil fmeidanisceliagdccunicampbr Partially supp orted by FAPESP and CNPq On the edgecolouring of split graphs in In that work we discussed lo cal sucient conditions for a graph to b e Class We considered sub classes of chordal graphs such as split graphs and interval graphs We b egin this note characterizing two sub classes of split graphs which we call easy and difficult The easy ones can b e easily coloured by a pullback technique We essentially b orrow the colour from a complete graph The difficult ones as the name indicates are hard to colour but this class has the imp ortant prop erty that if all its memb ers are Class then al l o dd maximum degree split graphs are Class Next we show that every difficult graph can b e transformed into an easy one by rep eated applications of a sub division op eration on vertices The idea is to use this trans formation to transfer the colouring of the easy graph to the difficult one The transfor mation increases the numb er of vertices but keeps the numb er of edges constant We were able to colour some difficult graphs with this technique and we show these results in the sequel Sp ecically we decided the cases where one sub division op eration is enough Denitions and notations General terms In this pap er G denotes a simple undirected nite connected graph V G and E G are the vertex and edge sets of G A stable set is a set of vertices pairwise nonadjacent in G A clique is a set of vertices pairwise adjacent in G A maximal clique of G is a clique not prop erly contained in any other clique A maximum clique of G is a clique of maximum size ie a clique with the largest p ossible numb er of vertices A subgraph of G is a graph H with V H V G and E H E G For X V G we denote by GX the subgraph induced by X that is V GX X and E GX consists of those edges of E G having b oth ends in X For each vertex v of a graph G Adj v denotes the set of vertices that are adjacent to v In addition N v denotes the neighbourhood of v that is N v Adj v fv g The degree of a vertex v is deg v jAdjv j The maximum degree of a graph G is then G max deg v A vertex u is universal if deg u jV Gj A vertex u is quasi v V (G) k universal if deg u jV Gj Given a graph G and k we denote by G the graph k k having V G V G and satisfying xy E G if and only if x and y are distinct and their distance in G is at most k The diameter of G is diamG max distv w v w V (G) K denotes the complete graph on n vertices n A split graph is a graph whose vertex set admits a partition into a stable set and a clique Colouring An assignment of colours to the vertices of G is a function V G S The elements of the set S are called colours A conict in an assignment of colours is the existence of two adjacent vertices with the same colour A vertexcolouring of a graph is an assignment of colours such that there are no conicts The chromatic number of a graph G is the minimum numb er of colours used among all vertexcolourings of G and is denoted by G On the edgecolouring of split graphs An assignment of colours to the edges of G is a function E G S Again the elements of the set S are called colours A conict in an assignment of colours is the existence of two edges with the same colour incident to a common vertex A vertex u is said to b e satised when uv uw implies v w for all neighb ours v w of u An edgecolouring of a graph is an assignment of colours such that every vertex is satised or equivalently such that there are no conicts The chromatic index of a graph G is the minimum numb er of colours used among all edgecolourings of G and is denoted by G A graph G is said to b e Class if G G and Class if G G Vizings theorem states that there are no other p ossibilities all graphs are either Class or Class A constructive pro of of Vizings theorem app eared in easy and difficult Recall that split graph is a graph G for which V G can b e partitioned into two sets A and B such that A is a clique and B is a stable set A splitting clique A of G is a maximal clique such that V G n A is a stable set of G Note that every splitting clique is a clique of maximum size in G We concentrate on split graphs of o dd maximum degree as mentioned in the intro duction We now dene a sub class of o dd maximum degree split graphs called easy which contains all graphs we consider easy to colour with colours An o dd maximum degree 2 split graph G is said to b e easy if G G Theorem easy Class The pro of is based on the concept of a pullback function A pul lback from graph G to graph H is a function f V G V H such that f is a homomorphism ie f takes adjacent vertices of G into adjacent vertices of H f is injective when restricted to each neighb ourho o d The following results relate edgecolouring to pullbacks They are proved in Theorem If f is a pul lback from G to H and H is k edgecolourable then G is k edge colourable Theorem Composition of pul lbacks is a pul lback 2 Any graph satisfying G G admits a pullback into the complete graph K In particular easy graphs can b e coloured with colours b ecause they admit a +1 pullback into the graph K a complete graph with an even numb er of vertices +1 We now dene the sub class of difficult graphs An o dd maximum degree split graph G is said to b e difficult when all vertices in a splitting clique A of G have degree G diamG On the edgecolouring of split graphs The imp ortance of this class is justied by Theorem b elow Together with Theorem they imply that if we prove that difficult Class then all o dd maximum degree split graphs are Class Theorem If G is any odd maximum degree split graph then there is a difficult graph H with G H and a pul lback function f V G V H Pro of Supp ose G is a split graph with diamG Let V G b e partitioned into a splitting clique A and a stable set B If there is a vertex v in A with deg v then we add an edge b etween v and any vertex u not adjacent to v This addition do es not change the maximum degree b ecause deg u If u now sees every vertex of A then A A fug is now a splitting clique with corresp onding stable set B B n fug By rep eating this op eration as many times as necessary we eventually obtain a graph satisfying the rst condition in the denition of difficult If at this p oint the diameter is greater than we take any two vertices x and y with distx y and join them into a single vertex z If vertex z sees every vertex of A then A A fz g is now a splitting clique with corresp onding stable set B B n fz g If necessary we add edges incident to z until we obtain a graph satisfying the rst condition in the denition
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