On the classification problem for split graphs

The Classification Problem is the problem of deciding whether a simple graph has chromatic index equal to Δ or Δ+1. In the first case, the graphs are called Class 1, otherwise, they are Class 2. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. Split graphs are a subclass of chordal graphs. Figueiredo at al. (J. Combin. Math. Combin. Comput. 32:79–91, 2000) state that a chordal graph is Class 2 if and only if it is neighborhood-overfull. In this paper, we give a characterization of neighborhood-overfull split graphs and we show that the above conjecture is true for some split graphs.

An easy lower bound for the chromatic index is the maximum vertex degree . A celebrated theorem of Vizing [27] states that, for a simple graph, the chromatic index is at most + 1. It was the origin of the Classification Problem that consists of deciding whether a given graph has chromatic index equal to or + 1. Graphs whose chromatic index is equal to are said to be Class 1; graphs whose chromatic index is equal to + 1 are said to be Class 2. Despite the restriction imposed by Vizing, it is NP-complete to determine, in general, if a graph is Class 1 [15]. In 1991, Cai and Ellis [2] proved that this holds also when the problem is restricted to some classes of graphs such as perfect graphs. However, the classification problem is entirely solved for some well-known classes of graphs that include the complete graphs, bipartite graphs [16], complete multipartite graphs [14], and graphs with universal vertices [20]. Nevertheless, the complexity of the classification problem is unknown for several well-studied strongly structured graph classes such as cographs [1], join graphs [17,24,25], planar graphs [23], chordal graphs, and several subclasses of chordal graphs such as split graphs [3], indifference graphs, interval graphs, and doubly chordal graphs [9]. By Vizing's theorem, to show that a graph G is Class 1 it is enough to construct an edge-coloring for G with (G) colors, however, to show that G is Class 2 we must prove that G does not have an edge-coloring with (G) colors. Considering a simple graph G, the inequality |E(G)| > (G) |V (G)| 2 is a useful sufficient condition to classify G as a Class 2 graph. In such a way, this condition implies that G has "many edges" and it is called overfull graph. Note that if a graph G is overfull, then G has an odd number of vertices and, since at most |V (G)| 2 edges of G can be colored with the same color, it is Class 2. Moreover, if a graph G has an overfull subgraph H with (H ) = (G), it is a subgraph-overfull graph [11]. When the overfull sub-graph H is induced by a (G)-vertex v and all its neighbors, denoted by N [v], we say that G is a neighborhood-overfull graph [8]. Overfull, subgraph-overfull, and neighborhoodoverfull graphs are Class 2. Although very rare, there are examples of Class 2 graphs that are neither subgraph-overfull nor neighborhood-overfull. The smallest one is P * , the graph obtained from the Petersen graph by removing an arbitrary vertex.
Hilton and Chetwynd [4] conjectured that being Class 2 is equivalent to being subgraph-overfull, when the graph has a maximum degree greater than |V (G)| 3 . This conjecture is known as the Overfull Conjecture. Every Class 2 graph with maximum degree at least |V (G)|−3 is subgraphoverfull [5]; every Class 2 complete multipartite graph is overfull [14]. These classes provide evidence for the Overfull Conjecture. Note that if the Overfull Conjecture is true, the resulting theorem can not be improved, since |V (P * )| 3 = (P * ). A split graph is a graph whose vertex set admits a partition into a clique and a stable set. Split graphs are a well-studied class of graphs for which most combinatorial problems are solved [6,7,18,19,22]. It has been proved that every overfull split graph contains a universal vertex and, therefore, is neighborhood-overfull. Moreover, every subgraph-overfull split graph is in fact neighborhoodoverfull [8]. In the same article, the authors have posed the following conjecture for chordal graphs (graphs without induced cycles C n with n ≥ 4), a superclass of split graphs.

Conjecture 1 Every Class 2 chordal graph is neighborhood-overfull.
Note that the validity of this conjecture for chordal graphs and, therefore, for split graphs implies that the edge-coloring problem for the corresponding class is in P .
In this work, we present a structural characterization of the neighborhood-overfull split graphs. If Conjecture 1 is true for split graphs, we are presenting a structural characterization of the unique Class 2 split graphs.
The study of the core and the semi-core of a graph gives us some information about the Classification Problem. The core of a graph G, denoted by G , is the subgraph of G induced by the (G)-vertices. The core of a graph has been studied, since 1965, when Vizing [28] proved that G is Class 1 if G has at most two vertices. This result was later generalized by Fournier [10]: if G is a forest, then G is Class 1. Thus, the question was what happens when the core of a graph contains a cycle. Hilton and Zhao [12,13] considered the graphs whose core is the disjoint union of cycles and paths, i.e., (G ) = 2 and they conjectured that every graph with (G ) = 2, different from P * , is Class 2 if and only if it is overfull. Tan and Hung [26] proved that this conjecture is true for split graphs. Note that a split graph with (G ) = 2 has 3 vertices with maximum degree.
The semicore of a graph G is the subgraph induced by the core of G and their neighbors. An interesting result says that the chromatic index of a graph is equal to the chromatic index of its semicore [17]. In general, to solve the Classification Problem for the semicore of G is as hard as to solve the Classification Problem for G. However, under special conditions of the semicore, a useful tool to solve the Classification Problem for G is to solve it for its semicore. We use this approach to classify some split graphs.
In Sect. 2, we recall some known results that we use in the subsequent sections. In Sect. 3, we give a characterization of neighborhood-overfull split graphs, and in Sect. 4, we show that Conjecture 1 is true for some subclasses of split graphs.

Theoretical framework
In this paper, G denotes a simple, finite, undirected, and connected graph with vertex set V (G) and edge set E(G).
When there is no ambiguity, we remove the symbol G from the notation.
A clique is a set of pairwise adjacent vertices of a graph. A maximal clique is a clique that is not properly contained in any other clique. A stable set is a set of pairwise nonadjacent vertices. A split graph G = {Q, S} is a graph whose vertex set admits a partition {Q, S} into a clique Q and a stable set S.
In the following, we shall use some known results that we recall for reader's convenience.
Theorem 3 [17] The chromatic index of a graph G is equal to the chromatic index of its semicore.
is odd, then G is Class 1.

A Class 2 split graph
By Theorem 5, neighborhood-overfull and subgraph-overfull concepts are equivalent when restricts to split graphs.
In this section, we give a structural characterization of split graphs that are neighborhood-overfull. As far as we know, these graphs are the unique known Class 2 split graphs. From now on, we consider a split graph G = {Q, S}, where Q is a maximal clique and S is a stable set. We shall associate to G a bipartite graph B obtained from G by removing all edges of the subgraph of G induced by Q. .
Proof Let G = {Q, S} be a split graph. If (G) is odd, by Theorem 4, G is Class 1 and, therefore, G is not neighborhood-overfull. Hence, (G) = |Q| + d(Q) − 1 must be even. This implies that |Q| and d(Q) have different parities. Assume that G is neighborhood-overfull. If G is a complete graph, it is known that |Q| must be odd with |Q| ≥ 3 and the lemma follows. Therefore, we consider S = ∅.
Now we give a characterization of neighborhood-overfull split graphs. It is relevant to note that the next theorem guarantees that every neighborhood-overfull split graph G contains a minimum number of (G)-vertices that have the same neighborhood.
If G is a complete graph, every vertex is a (G)-vertex and all the conditions are trivially true. Therefore, we con- ] is overfull. Then condition (1) is true. Moreover, by Theorem 2, , all vertices in X are (G)-vertices and they are twins. Furthermore, (1) and (2)  2 − 1. By condition (1), G has even maximum degree. Therefore, G is a neighborhood-overfull graph.
The split graphs described in Theorem 9 are Class 2. Therefore, if the Conjecture 1 were true, these graphs would be the unique Class 2 split graphs and every split graph G = {Q, S} with (G) even and |Q| < (d(Q)) 2 + 3 would be Class 1.
The split graphs described in Theorem 9 are Class 2. Therefore, if the Conjecture 1 were true, these graphs would be the unique Class 2 split graphs and every split graph G = {Q, S} with (G) even and |Q| < (d(Q)) 2 + 3 would be Class 1.

Some Class 1 split graphs
In this section, we use the semi-core of a given split graph G to determine the chromatic index of G. In general, finding the chromatic index of the semicore of a graph G may be as difficult as finding the chromatic index of G. If G has the hereditary property for induced subgraphs, G and its semicore belongs to the same class. This is the case of split graphs. However, under special conditions this approach may be useful for classifying certain split graphs.

Theorem 11 Let G = {Q, S} be a split graph and let H be the semicore of G. If H has a universal vertex, then G is Class 2 if and only if H is overfull.
Proof Let G = {Q, S} be a split graph and let H be the semicore of G. Then H is also a split graph and (H ) = (G). To calculate the chromatic index of G, by Theorem 3, it is sufficient to calculate the chromatic index of its semicore. By the hypothesis, H has a universal vertex. So, by Theorem 2, H is Class 2 if and only if H is overfull. Therefore, G is Class 2 if and only if H is overfull.
By Theorem 11, the Classification Problem is solved for a split graph if its semicore has a universal vertex. Now, we consider split graphs whose semicore has no universal vertex.
The next theorem shows that the Classification Problem is also solved for split graphs G that have (G) even and (G) ≥ |Q| + |X| − 2, where X is the set of (G)-vertices. Now, we consider a split graph G = {Q, S} with |X| ≥ 4 vertices of maximum degree and an even (G) = |Q| + |X| − 3. Note that (G ) ≥ 3. (Recall that Tan and Hung [26] proved that when (G ) = 2, a split graph G is Class 2 if and only if G is overfull.) First, we show that, when (G ) ≥ 3 and (G) = |Q| + |X| − 3, G is not neighborhood-overfull. After we show that, if |X| is odd, then G is Class 1.

Lemma 14
Let G = {Q, S} be a split graph with even maximum degree. Let X be the set of (G)-vertices with |X| ≥ 4.

Lemma 15
Let G = {Q, S} be a split graph with even maximum degree. Let X be the set of (G)-vertices with |X| > 4. If (G) = |Q| + |X| − 3 and |X| is odd, then G is Class 1.
Proof Let G = {Q, S} be a split graph with even maximum degree. Let X be the set of (G)-vertices with odd |X| > 4. Since (G) = |Q| + |X| − 3, d(Q) = |X| − 2. By Lemma 14, G is not neighborhood-overfull and we shall prove that G is Class 1.
To calculate the chromatic index of G, by Lemma 3, it is sufficient to calculate the chromatic index of its semicore. Let X = {v 0 , . . . , v |X|−1 } be the set of (G)-vertices. Since We use the algorithm of Chen, Fu, and Ko, with a ( − 1) × ( − 1) Latin square LS = (m ij ) defined by m ij = i + j (mod − 1), 0 ≤ i, j ≤ − 2, to give an edge-coloring π for L with − 1 colors.
In order to give this edge-coloring for L, they use a special ordering of the vertices of L[Q] constructed as follows. Consider an ordering (u 0 , u 1 , . . . , u |S L |−1 ) of the vertices of S L such that N L (y) ∩ S L are the first ones and u is the last one. Let U = (u 0 , u 1 , . . . , u h ), where h is the minimum number with |N(U )| ≥ |Q|/2. Let B be the bipartite subgraph of L with partition {Q, U } and edges with an end in Q and another one in U . A CFK-ordering is an ordering of the vertices of Q such that d B (v i We show that there exists a CFK-ordering of the vertices of Q such that y and z are the first ones in the ordering. By definition of Latin square LS and considering the coloring applied to L[Q], we can observe that each edge v i v j ∈ Q such that i + j = |Q| − 1 (mod − 1) is colored |Q| − 1. Since |X| is odd, is even, and = |Q| + |X| − 3, |Q| is even. Since |Q| is even, each edge incident to a vertex of S L has a color distinct from |Q| − 1. Now, we assign colors to each edge vw of H . For this, consider the vertices of H in the same order they are in L. We must distinguish two cases depending on whether v |Q|−1 is a -vertex or not. Note that in the first case there exists an The colors are assigned by π , for each vw ∈ E(G), as follows.
Since |Q| is even, no edge incident to a vertex of S H is colored |Q| − 1. Since u = u , we can color the edge v 0 u and the edge v |Q|−1 u , if it exists, with color |Q| − 1.
Therefore, H has an edge-coloring with colors and, by Lemma 3, G is Class 1.

Conclusions
The split graphs described in Theorem 9 are Class 2. If the Conjecture 1 were true, then the graphs described in Theorem 9 would be the unique Class 2 split graphs.
By Corollary 10, every neighborhood-overfull split graph has (G) ≥ |V (G)| 3 . So, if Conjecture Overfull were true for split graphs, it provides a strong evidence that Conjecture 1 were true for these graphs.
Theorem 16 gives another evidence that Conjecture 1 is true, providing a polynomial algorithm for edge-coloring with colors the graphs of a new set of non neighborhoodoverfull split graphs. It is known that if G is a graph with (G) ≥ |V (G)| − 3, then G is Class 1 if and only if G is not subgraph-overfull [5,20,21]. For a split graph G with (G) even, the condition on (G), given by Theorem 16, can be smaller than |V (G)| − 3.