All our graphs are finite and have no loops or multiple edges. For any integer , a -coloring of a graph is a mapping such that any two adjacent vertices in satisfy . A graph is -colorable if it admits a -coloring. The chromatic number of a graph is the smallest integer such that is -colorable. In general, determining whether a graph is -colorable or not is well-known to be -complete for every fixed . Thus designing algorithms for computing the chromatic number by putting restrictions on the input graph and obtaining bounds for the chromatic number are of interest.
A clique in a graph is a set of pairwise adjacent vertices. Let denote the maximum clique size in a graph . Clearly for every induced subgraph of . A graph is perfect if every induced subgraph of satisfies . The existence of triangle-free graphs with aribtrarily large chromatic number shows that for general graphs the chromatic number cannot be upper bounded by a function of the clique number. However, for restricted classes of graphs such a function may exist. Gyárfás  called such classes of graphs -bounded classes. A family of graphs is -bounded with -bounding function if, for every induced subgraph of , . For instance, the class of perfect graphs is -bounded with .
Given a family of graphs , a graph is -free if no induced subgraph of is isomorphic to a member of ; when has only one element we say that is -free. Several classes of graphs defined by forbidding certain families of graphs were shown to be -bounded: even-hole-free graphs 
; odd-hole-free graphs; quasi-line graphs ; claw-free graphs with stability number at least 3 ; see also [6, 8, 12, 22, 24] for more instances.
For any integer we let denote the path on vertices and denote the cycle on vertices. A cycle on vertices is referred to as a square. It is well known that every -free graph is perfect. Gyárfás  showed that the class of -free graphs is -bounded. Gravier et al.  improved Gyárfás’s bound slightly by showing that every -free graph satisfies . In particular every -free graph satisfies . Improving this exponential bound seems to be a difficult open problem. In fact the problem of determining whether the class of -free graphs admits a polynomial -bounding function remains open, and the known -bounding function for such class of graphs satisfies . So the recent focus is on obtaining (linear) -bounding functions for some classes of -free graphs, where . It is shown in  that every -free graph satisfies , and in  that every -free graph satisfies . Gaspers and Huang  studied the class of -free graphs (which generalizes the class of -free graphs and the class of -free graphs) and showed that every such graph satisfies . We improve their result and establish the best possible bound, as follows.
Let be any -free graph. Then . Moreover, this bound is tight.
The degree of a vertex in is the number of vertices adjacent to it. The maximum degree over all vertices in is denoted by . For any graph , we have . Brooks  showed that if is a graph with and , then . Reed  conjectured that every graph satisfies . Despite several partial results [25, 31, 33], Reed’s conjecture is still open in general, even for triangle-free graphs. Using Theorem 1.1, we will show that Reed’s conjecture holds for the class of (,)-free graphs:
If is a -free graph, then .
One can readily see that the bounds in Theorem 1.1 and in Theorem 1.2 are tight on the following example. Let be a graph whose vertex-set is partitioned into five cliques such that for each , every vertex in is adjacent to every vertex in and to no vertex in , and for all (). Clearly and . Since has no stable set of size , is -free and . Moreover, since no two non-adjacent vertices in has a common neighbor in , we also see that is -free.
Finally, we also have the following result.
There is a polynomial-time algorithm which computes the chromatic number of any -free graph.
The proof of Theorem 1.3 is based on the concept of clique-width of a graph , which was defined in  as the minimum number of labels which are necessary to generate using a certain type of operations. (We omit the details.) It is known from [26, 32] that if a class of graphs has bounded clique-width, then there is a polynomial-time algorithm that computes the chromatic number of every graph in this class. We are able to prove that every -free graph that has no clique cutset has clique-width at most , which implies the validity of Theorem 1.3. However a similar result, using similar techniques, was proved by Gaspers, Huang and Paulusma . Hence we refer to , or to the extended version of our manuscript  for the detailed proof of Theorem 1.3.
We finish on this theme by noting that the class of -free graph itself does not have bounded clique-width, since the class of split graphs (which are all -free) does not have bounded clique-width [2, 29]. The clique-width argument might also be used for solving other optimization problems in (-free graphs, in particular the stability number. However this problem was solved earlier by Mosca , and the weighted version was solved in , and both algorithms have reasonably low complexity.
In a graph , the neighborhood of a vertex is the set ; we drop the subscript when there is no ambiguity. The closed neighborhood is the set . Two vertices are clones if . For any and , we let . For any two subsets and of , we denote by , the set of edges that has one end in and other end in . We say that is complete to or is complete if every vertex in is adjacent to every vertex in ; and is anticomplete to if . If is singleton, say , we simply write is complete (anticomplete) to instead of writing is complete (anticomplete) to . If , then denote the subgraph induced by in . A vertex is universal if it is adjacent to all other vertices. A stable set is a set of pairwise non-adjacent vertices. A clique-cutset of a graph is a clique in such that has more connected components than . A matching is a set of pairwise non-adjacent edges. The union of two vertex-disjoint graphs and is the graph with vertex-set and edge-set . The union of copies of the same graph will be denoted by ; for example denotes the graph that consists in two disjoint copies of .
A vertex is simplicial if its neighborhood is a clique. It is easy to see that in any graph that has a simplicial vertex, letting denote the set of simplicial vertices, every component of is a clique, and any two adjacent simplicial vertices are clones.
A hole is an induced cycle of length at least . A graph is chordal if it contains no hole as an induced subgraph. Chordal graphs have many interesting properties (see e.g. ), in particular: every chordal graph has a simplicial vertex; every chordal graph that is not a clique has a clique-cutset; and every chordal graph that is not a clique has two non-adjacent simplicial vertices.
In a graph , let be disjoint subsets of . It is easy to see that the following two conditions (i) and (ii) are equivalent: (i) any two vertices satisfy either or ; (ii) any two vertices satisfy either or . If this condition holds we say that the pair is graded. Clearly in a -free graph any two disjoint cliques form a graded pair. See also Lemma 2.3 below.
Some special graphs
Let be five graphs, as shown in Figure 2, where is the Petersen graph.
For integers let be the graph whose vertex-set can be partitioned into sets and such that:
is a clique of size , and is a stable set of size , and the edges between and form a matching of size , namely, ;
is a clique of size , and is a stable set of size , and the edges between and form a matching of size , namely, ;
The neighborhood of is ;
The neighborhood of is ;
The neighborhood of is .
See Figure 3 for the schematic representation of the graph and for the graph .
A blowup of a graph is any graph such that can be partitioned into (not necessarily non-empty) cliques , , such that is complete if , and if . See Figure 4:(a) for a blowup of a .
A band is any graph (see Figure 4:(b)) whose vertex-set can be partitioned into seven sets such that:
Each of is a clique.
The sets , , and are complete.
The sets , and are empty.
The pairs , and are graded.
A belt is any -free graph (see Figure 4:(c)) whose vertex-set can be partitioned into seven sets such that:
Each of is a clique.
The sets and are complete.
The sets , , are empty.
For each , is complete, every vertex in has a neighbor in , and no vertex of is universal in .
A boiler is a -free graph whose vertex-set can be partitioned into five sets such that:
The sets , , and are non-empty, and , and are cliques.
The sets , , and are complete.
The sets , and are empty.
and are -free.
Every vertex in has a neighbor in .
For some integer , is partitioned into non-empty sets , pairwise anticomplete, and is partitioned into non-empty sets , such that for each every vertex in has a neighbor in and no neighbor in ; and every vertex in has a neighbor in .
is complete, and for each every vertex in is either complete or anticomplete to , and no vertex in is complete to .
See Figure 5 for the partial structure of a boiler.
We consider that the definition of blowups (of certain fixed graphs) and of bands (using Lemma 2.3) is also a complete description of the structure of such graphs. However this is not so for belts and boilers. Such graphs have additional properties, and a description of their structure is given in Section 4.
Now we can state our main structural result. The existence of such a decomposition theorem was inspired to us by the results from  which go a long way in that direction.
If is any -free graph, then one of the following holds:
has a clique cutset.
has a universal vertex.
is a blowup of either , or (for some ).
is either a band, a belt, or a boiler.
Let be a -free graph that has no clique-cutset and no universal vertex. Then the following hold:
If contains an , then is a blowup of .
If contains an and no , then is a band.
If is -free, and contains an induced , then is a blowup of one of the graphs .
If is -free, and contains an , then is a blowup of either or for some integers .
If contains no and no , and contains a , then is either a belt or a boiler.
Proof of Theorem 1.4, assuming
Let be any -free graph. If is chordal, then either is a complete graph (so it has a universal vertex) or has a clique cutset. Now suppose that is not chordal. Then it contains an induced cycle of length either or . So it satisfies the hypothesis of one of the items of Theorem 1.5 and consequently it satisfies the conclusion of this item. This established Theorem 1.4.
2 Classes of square-free graphs
In this section, we study some classes of square-free graphs and prove some useful lemmas and theorems that are needed for the later sections. We first note that any blowup of a -free chordal graph is -free chordal.
In a chordal graph , every non-simplicial vertex lies on a chordless path between two simplicial vertices.
Proof. Let be a non-simplicial vertex in , so it has two non-adjacent neighbors . If both are simplicial, then -- is the desired path. Hence assume that is non-simplicial. Since is not a clique, it has two simplicial vertices, so it has a simplicial vertex different from . So . In , the vertex is non-simplicial, so, by induction, there is a chordless path --- in , with , such that and are simplicial in and for some . If and are simplicial in , then is the desired path. So suppose that is not simplicial in , so . Since is simplicial in we have . Then we see that either ---- or --- is the desired path.
In a chordal graph , let and be disjoint subsets of such that is a clique and every simplicial vertex of has a neighbor in . Then every vertex in has a neighbor in .
Proof. Consider any non-simplicial vertex of . By Lemma 2.1 there is a chordless path --- in , with , such that and are simplicial in and for some . By the hypothesis has neighbor and has a neighbor in . Suppose that has no neighbor in . Let be the largest integer in such that has a neighbor in , and let be the smallest integer in such that has a neighbor in . Then contains a hole, a contradiction. So has a neighbor in .
In a -free graph , let be two disjoint cliques. Then:
There is a labeling of the vertices of such that . Similarly, there is a labeling of the vertices of such that .
If every vertex in has a neighbor in , then some vertex in is complete to .
If every vertex in has a non-neighbor in , then some vertex in is anticomplete to .
If is not complete, there are indices and such , and for all , and for all . Moreover, every maximal clique of contains one of .
Proof. Consider any two vertices . If there are vertices and , then induces a . Hence we have either or . This inclusion relation for all implies the existence of a total ordering on , which corresponds to a labeling as desired, and the same holds for . This proves the first item of the lemma. The second and third item are immediate consequences of the first.
Now suppose that is not complete to . Consider any vertex that has a non-neighbor in , and let be the smallest index such that . Let be the smallest index such that . So . We have for all by the choice of . We also have for all , for otherwise, since we also have , contradicting the definition of . This proves the first part of the fourth item.
Finally, consider any maximal clique of . Let be the largest index such that and let be the largest index such that . By the properties of the labelings and the maximality of we have . If both and , then the properties of imply that (and also ) is a clique of , contradicting the maximality of . Hence we have either or , and so contains one of .
In a -free graph , let , and be disjoint subsets of such that:
is a clique, and every vertex in has a neighbor in ,
is complete to and anticomplete to ;
Either is not connected, or there are vertices such that is complete to and anticomplete to , and is anticomplete to , and .
Then is -free.
Proof. First suppose that there is a --- in . By the hypothesis has a neighbor . Then , for otherwise induces a ; and similarly . If is connected, then either ----- or ----- is a . Now suppose that is not connected. So contains a vertex that is anticomplete to . By the hypothesis has a neighbor . As above we have and for all for otherwise there is a . But then either ----- or ----- is a .
Now suppose that there is a in , with vertices and edges , . We know that has a neighbor , and as above we have for each , for otherwise there is a . Likewise, has a neighbor , and for each . Then ----- is an induced for some and .
We want to understand the structure of -free graphs
as they play a major role in the structure of belts and boilers.
Recall that -free graphs were studied by Golumbic
, who called them trivially perfect graphs.
Clearly any such graph is chordal. It was proved in  that
every connected )-free graph has a universal vertex. It
follows that trivially perfect graphs are exactly the class
of graphs that can be built recursively as follows, starting from
– The disjoint union of any number of trivially perfect graphs is trivially perfect;
– If is any trivially perfect graph, then the graph obtained from by adding a universal vertex is trivially perfect.
As a consequence, any connected member of can be represented by a rooted directed tree defined as follows. If is a clique, let have one node, which is the set . If is not a clique, then by Golumbic’s result the set of universal vertices of is not empty, and has a number of components . Let then be the tree whose root is and the children (out-neighbors) of are the roots of .
The following properties of appear immediately. Every node of is a non-empty clique of , and every vertex of is in exactly one such clique, which we call ; moreover, is a homogeneous set (all member of are pairwise clones). For every vertex of , the closed neighborhood of consists of and all the vertices in the cliques that are descendants and ancestors of in . Every maximal clique of is the union of the nodes of a directed path in . All vertices in any leaf of are simplicial vertices of , and every simplicial vertex of is in some leaf of .
We say that a member of is basic if every node of is a clique of size . (We can view as a directed tree, where every edge is directed away from the root; and then is the underlying undirected graph of the transitive closure of .). It follows that every member of is a blowup of a basic member of . In a basic member of , two vertices are adjacent if and only if one of them is an ancestor of the other in , and every clique of consists of the set of vertices of any directed path in .
A dart is the graph with vertex-set and edge-set . Let be the tree obtained from by subdividing one edge. Next we give the following useful lemma.
Let be a -free graph.
(a) If does not have three pairwise non-adjacent simplicial vertices, then is a blowup of .
(b) If does not have four pairwise non-adjacent simplicial vertices, then is a blowup of a dart.
Proof. The hypothesis of (a) or (b) means that, if is a connected component of , then is a tree with at most three leaves. Since each internal vertex of has at least two leaves, is either , , (rooted at its vertex of degree ), (rooted at its vertex of degree ), or (rooted at its vertex of degree ). Then the conclusion follows directly from our assumption on and the preceding arguments.
Let be the class of -free graphs. So . If is any member of , and is connected and not a clique, then since is -free all components of , except possibly one, are cliques. So all children of in , except possibly one, are leaves. Applying this argument recursively we see that the tree consists of a rooted directed path plus a positive number of leaves adjacent to every node of this path, with at least two leaves adjacent to the last node of this path. We call such a tree a bamboo. By the same argument as above, every member of is a blowup of a basic member of .
A graph is a -pair if is -free, chordal, and can be partitioned into two sets and such that is a clique, , every vertex in has a neighbor in , and any two non-adjacent vertices in have no common neighbor in . Depending on the context we may also write that is a -pair.
We say that is a basic -pair if the subgraph is a basic member of , with vertices for some integer , and a clique ; and for each , if is simplicial in then , else consists of plus the union of over all descendants of in .
Before describing how all -pairs can be obtained from basic -pairs we need to introduce another definition. Let be any graph and be a matching in . An augmentation of along is any graph whose vertex-set can be partitioned into cliques , , such that is complete if , and if , and is a graded pair if . (See  for a similar definition.)
In a basic -pair , with the same notation as above, we say that a matching is acceptable if there is a clique in such that .
A graph is a -pair then it is an augmentation of a basic -pair along an acceptable matching.
Proof. Let be any -pair, with the same notation as above. Since is -free it admits a representative tree which is a bamboo. We claim that:
|If are two nodes of such that is a descendant of , then is complete to .||(1)|
Proof: Consider any and ; so there is a vertex with . Since is not a leaf of , there is a child of in such that is not on the directed path from to , and so and are not adjacent (they are anticomplete to each other). Pick any . Then and . We know that has a neighbor . We have by the definition of a -pair ( and have no common neighbor in ). Then , for otherwise contains an induced hole of length or , contradicting the fact that is chordal. So (1) holds.
Let be the nodes of . For each , let be the union of over all descendants of in , and let . Let (so ).
Let be the nodes of that are not homogeneous in (if any). Note that for each the pair is graded since is -free. We claim that:
|is a clique.||(2)|
Proof: Suppose, on the contrary, and up to symmetry, that is not complete, and so . For each , since is not homogeneous in , there are vertices and a vertex that is adjacent to and not to . Since non-adjacent vertices in have no common neighbor in , we have and . Then ----- is a . So (2) holds.
Let be the basic member of of which is a blowup. Let have vertices , where corresponds to the node of for all . Let be the graph obtained from by adding a set , disjoint from , and edges so that is a clique in and, for all and , vertices and are adjacent in if and only if in . By this construction and by (1) is a basic -pair. In let . It follows from (2) that is an acceptable matching of and from all the points above that is an augmentation of along .
3 Structure of (, )-free graphs
In this section, we give the proof of Theorem 1.5. We say that a subgraph of is dominating if every vertex in is a adjacent to a vertex in . We will use the following theorem of Brandstädt and Hoàng .
Theorem 3.1 ()
Let be a ()-free graph that has no clique cutset. Then
the following statements hold.
(i) Every induced is dominating.
(ii) If contains an induced which is not dominating, then is the join of a complete graph and a blowup of the Petersen graph.
In the next two theorems we make some general observations about the situation when a -free graph contains a hole (which must have length either or ). Observe that in a -free graph , if -- is a , then any which is adjacent to and is also adjacent to .
Let be any -free graph that contains a with vertex-set and . Let:
Moreover, let , , and . Then the following properties hold for all :
is a clique.
, , , , and are empty.