Abstract
An -coloring of a simple connected graph is an assignment of nonnegative integers to the vertices of such that if and if for all , where denotes the distance between and in . The span of is the maximum color assigned by . The span of a graph , denoted by , is the minimum of span over all -colorings on . An -coloring of with span is called a span coloring of . An -coloring is said to be irreducible if there exists no -coloring g such that for all and for some . If is an -coloring with span , then is a hole if there is no such that . The maximum number of holes over all irreducible span colorings of is denoted by . A tree with maximum degree having span is referred to as Type-I tree; otherwise it is Type-II. In this paper, we give a method to construct infinitely many trees with at least one hole from a one-hole tree and infinitely many two-hole trees from a two-hole tree. Also, using the method, we construct infinitely many Type-II trees with maximum number of holes one and two. Further, we give a sufficient condition for a Type-II tree with maximum number of holes zero.
1. Introduction
The channel assignment problem is the problem of assigning frequencies to transmitters in some optimal manner. In 1992, Griggs and Yeh [1] have introduced the concept of -coloring as a variation of channel assignment problem. The distance between two vertices and in a graph , denoted by , is defined as the length of a shortest path between and in . An -coloring of a graph is an assignment such that, for every in , if and are adjacent and if and are at distance 2. The nonnegative integers assigned to the vertices are also called colors. The span of , denoted by , is . The span of , denoted by , is . An -coloring with span is called a span coloring. A tree is a connected acyclic graph. In the introductory paper, Griggs and Yeh [1] proved that for ; is either or for any tree with maximum degree . We refer to a tree as Type-I if ; otherwise it is Type-II. In a graph with maximum degree , we refer to a vertex as a major vertex if its degree is ; otherwise is a minor vertex. Wang [2] has proved that a tree with no pair of major vertices at distances 1, 2, and 4 is Type-I. Zhai et al. [3] have improved the above condition as a tree with no pair of major vertices at distances 2 and 4 is Type-I. Mandal and Panigrahi [4] have proved that if has at most one pair of major vertices at distance either 2 or 4 and all other pairs are at distance at least 7. Wood and Jacob [5] have given a complete characterization of the -span of trees up to twenty vertices.
Fishburn and Roberts [6] have introduced the concept of no-hole -coloring of a graph. If is an -coloring of a graph with span , then an integer is called a hole in if there is no vertex in such that . An -coloring with no hole is called a no-hole coloring of . Fishburn et al. [7] have introduced the concept of irreducibility of -coloring. An -coloring of a graph is reducible if there exists another -coloring of such that for all vertices in and there exists a vertex in such that . If is not reducible then it is called irreducible. An irreducible no-hole coloring is referred to as inh-coloring. A graph is inh-colorable if there exists an inh-coloring. For an inh-colorable graph , the lower inh-span or simply inh-span of , denoted by , is defined as . Fishburn et al. [7] have proved that paths, cycles, and trees are inh-colorable except , , and stars. In addition to that, they showed that where is any nonstar tree. Laskar et al. [8] have proved that any nonstar tree is inh-colorable and . The maximum number of holes over all irreducible span colorings of is denoted by . Laskar and Eyabi [9] have determined the exact values for maximum number of holes for paths, cycles, stars, and complete bipartite graphs as 2, 2, 1, and 1, respectively, and conjectured that, for any tree , if and only if is a path . S. R. Kola et al. [10] have disproved the conjecture by giving a two-hole irreducible span coloring for a Type-II tree other than path.
In this article, we give a method of construction of infinitely many two-hole trees from a two-hole tree and infinitely many trees with at least one hole from a one-hole tree. Also, we find maximum number of holes for some Type-II trees given by Wood and Jacob [5] and obtain infinitely many Type-II trees of holes one and two by applying the method of construction. Further, we give a sufficient condition for a zero-hole Type-II tree.
2. Construction of Trees with Maximum Number of Holes One and Two
We start this section with a lemma which gives the possible colors to the major vertices in a two-hole span coloring of a Type-II tree.
Lemma 1. In any two-hole span coloring of a Type-II tree with , all major vertices receive either the same color or the colors from any one of the sets , , or .
Proof. Let be a two-hole span coloring of a Type-II tree . Suppose that and are major vertices such that . First, we prove that or or . Let and . Without loss of generality, we assume that . If , then the color 1 must be one of the two holes in . If , then and are the holes. Since cannot be 1, is 1 which implies . If , then and are the holes in . If , then and are the holes which are not possible as . If , then must be one of the holes in . Since cannot be , is which implies .
If , then 1 and 3 are the holes. If any major vertex receives a color other than 0 and 2, then the neighbors of cannot get the colors 1 and 3 and at least one of and . This is not possible as we need number of colors to color a major vertex and its neighbors. Similarly, other cases can be proved.
The following lemma is a direct implication of Lemma 1.
Lemma 2. If is a two-hole span coloring of a Type-II tree having two major vertices at distance less than or equal to two, then the set of holes in is , , or .
When we say connecting two trees, we mean adding an edge between them. Corresponding to the possibilities of holes given in Lemma 2, we give a list of trees which can be connected to a two-hole tree having two major vertices at distance less than or equal to two, to obtain infinitely many two-hole trees. Later, we give a list of trees which can be connected to a one-hole tree to get infinitely many one-hole trees.
Theorem 3. If is a tree with maximum number of holes two and having at least two major vertices at distance at most two, then there are infinitely many trees with maximum number of holes two and with maximum degree same as that of .
Proof. Let be an irreducible span coloring of with two holes. Then by Lemma 2, the set of holes in is or or . Now, we give a method to construct trees from using the coloring and holes in . For all the three possibilities of holes, we give a list of trees which can be connected to to get a bigger tree with maximum number of holes two. Suppose and are the holes in . We use Table 1 for construction.
Let be a vertex of the tree and be the color received by . Now depending on the colors of the neighbors of , to preserve -coloring, we connect the trees (one at a time) given in Table 1 by adding an edge between of and the vertex colored of tree in the table. Note that and the color is not equal to any of the colors , , , , and and not assigned to any neighbor of . To maintain irreducibility, we use the condition given in the last column of the table. It is easy to see that, after every step, we get a tree with maximum degree same as that of and a two-hole irreducible span coloring of . Also, it is clear that is a subtree of . Since connecting a tree to any pendant vertex is always possible, we get infinitely many trees.
Suppose and are the holes in . Construction is similar to the previous case using trees in Table 2.
Suppose and are the holes in . We use trees in Table 3 for construction.
Theorem 4. If T is a tree with , then there exist infinitely many trees containing and with maximum number of holes at least .
Proof. Here, we start with a one-hole irreducible -span coloring of having hole . The construction of infinitely many trees is similar to that in Theorem 3 and using Table 4. Since after every step we get a tree with one-hole irreducible span coloring, .
Theorem 5. If is a tree with and has no two-hole span coloring, then there exist infinitely many trees with maximum number of holes one and containing .
Proof. Since has no two-hole span coloring, any tree containing having same maximum degree as that of cannot have a two-hole span coloring. Therefore, every tree obtained from using Theorem 4 has maximum number of holes one.
Corollary 6. If is a Type-I tree and , then there exist infinitely many trees with maximum number of holes one and containing .
3. Maximum Number of Holes in Some Type-II Trees
Recall that, in a graph with maximum degree , we refer a vertex as a major vertex if its degree is . Otherwise is a minor vertex. Wood and Jacob [5] have given some sufficient conditions for a tree to be Type-II. We consider some of their sufficient conditions as below.
Theorem 7 (see [5]). A tree containing any of the following subtrees is Type-II provided the maximum degrees of the subtree and the tree are the same .(I): a tree with an induced consisting of three major vertices.(II): a tree with a minor vertex and at least 3 major vertices adjacent to .(III): a tree with a major vertex and at least major vertices at distance two from , and is not a subtree of the tree.(IV): a tree with a vertex adjacent to vertices and two neighbors of each , are major.
Since the above trees can be as small as possible, we consider the degrees of minor vertices as minimum as possible. Now, we find the maximum number of holes for the trees , and . For any tree with maximum degree , it is clear that . First, we show that , . Also, if has a vertex adjacent to at least four major vertices. Further, we give a two-hole -irreducible span coloring of if it has exactly major vertices at distance two from a major vertex and we show that , if has exactly major vertices at distance two from a major vertex. Later, we show that these upper bounds are the exact values by defining -irreducible span colorings with appropriate holes. Now onwards, unless we mention, tree refers to Type-II tree. In figures, we use symbol ▲ to denote a major vertex.
Theorem 8. For the trees , , .
Proof. Let , and be the major vertices of . Since , and receive three different colors in any -coloring, by Lemma 1, cannot have a two-hole irreducible span coloring. Similarly, we can prove that .
Now, we consider with labelling as in Figure 1.
Suppose that is a two-hole irreducible span coloring of . Then by Lemma 1, all major vertices of receive colors from or or . Suppose the major vertices receive 0 and 2. Then 1 and 3 are holes. Without loss of generality, we assume that and . Now, one of the pendant vertices adjacent to must receive a color grater than 3 which reduces to 3 giving a contradiction to the fact that is irreducible. Similarly, we can prove the other two cases. Therefore, .
Theorem 9. If at least four major vertices are adjacent to in , then .
Proof. Recall that is a tree with a vertex adjacent to at least three major vertices. Let , and be four major vertices adjacent to in . Suppose that it has a one-hole irreducible -span coloring . Let , and be the colors received by , and , respectively. Without loss of generality, we assume that . If , then except and all other colors are used to the neighbors of . Also, except and , all other colors are used to the neighbors of . Since , and are four different colors, cannot have a hole which is a contradiction. So, . Since is one-hole coloring, the colors , and cannot be four different colors and hence is the hole. Now, a pendant neighbor of receives which reduces to the hole giving a contradiction to the fact that is irreducible.
S. R. Kola et al. [10] have disproved the conjecture given by Laskar and Eyabi [9] by giving two-hole irreducible span colorings for Type-II trees of maximum degrees three and four. Following theorem gives a two-hole irreducible span coloring for a tree with maximum degree which is also a counterexample for the conjecture.
Theorem 10. If exactly major vertices are at distance two from the major vertex in , then .
Proof. Let be the tree with exactly major vertices at distance two to . It is easy to see that the -span coloring of the tree given in Figure 2 is irreducible with and as holes.
Theorem 11. If is the tree with exactly major vertices at distance two to , then .
Proof. We consider with labelling as in Figure 3.
Suppose that has an -span coloring with two holes. Then by Lemma 1, all major vertices of receive colors from or or . Suppose that assigns 0 and 2 to the major vertices. Since , , it is not possible to color all s as the four colors 0, 1, 2, and 3 cannot be assigned. Therefore, in this case, two-hole span coloring is not possible for . Similarly, we can prove the other two cases. Hence, .
Let be the tree with exactly three major vertices are adjacent to a vertex.
Theorem 12. For the trees , , , and , the maximum number of holes is one.
Proof. It is easy to see that the colorings of , , , and given in Figures 4, 5, 6, and 7, respectively, are irreducible -span colorings with hole .
4. Infinitely Many Trees with Holes 0, 1, and 2
Recall that is the tree with exactly three major vertices adjacent to a vertex and is the tree with exactly major vertices at distance two from a major vertex. Let be the tree with exactly four major vertices adjacent to the vertex . In this section, we give a sufficient condition for a Type-II tree to be a zero-hole tree. Also, we construct infinitely many trees with maximum number of holes 1 from each of the trees , , , , and and infinitely many two-hole trees containing .
Theorem 13. If the tree with at least five major vertices is a subtree of a tree with maximum degree same as that of , then .
Proof. Let , and be five major vertices adjacent to and receive the colors , and , respectively, by a one-hole span coloring of . Without loss of generality, we assume that . As in the proof of Theorem 9, we get and is the hole. Since , we have . Since , must be used to a neighbor of which is a contradiction. So, any -span coloring of with at least five major vertices is a no-hole coloring. Therefore, if a tree contains with at least five major vertices and with maximum degree same as that of , then .
Theorem 14. There are infinitely many trees with maximum number of holes one and containing each of the trees , , , , and .
Proof. First, we prove that , , , , and cannot have two-hole span coloring. From Theorems 8 and 11, it is clear that , , , and cannot have two-hole span colorings.
Next, we prove that cannot have a two-hole span coloring. We consider with the labelling as in Theorem 12. Suppose that has an -span coloring with two holes. By Lemma 1, any major vertex of receives the color from , , or . Suppose assigns 0 and 2 to major vertices. Then 1 and 3 are holes, , and cannot receive the colors , and . Therefore s receive different colors among ( in number) and so, one of these colors is not used, say . Since either or (if then ) is used to color one of the s, cannot be used to . Since 1 and 3 are holes, there is no color for . Similarly, we can prove the other cases.
Now, to use Theorem 4, we need one-hole irreducible span coloring of , , , , and . Since is 0, first we construct a tree from such that . We define a one-hole span coloring for as in Figure 8 ( is a subtree of ). Since the colors and received by the vertices adjacent to the vertex are reducible and there is no other color reducible, we connect star to the vertices to make the colors and irreducible. The tree obtained is .
Now, using Table 5 obtained from Table 4 corresponding to the hole and using irreducible one-hole span colorings of , , , and given in Theorem 12, we construct infinitely many one-hole trees containing each of the trees , , , and , respectively. We get infinitely many trees containing by using irreducible one-hole coloring of given in Figure 8 and using Table 5.
Example 15. In Figure 9, we illustrate the construction of one-hole tree as in Theorem 14 for the tree with maximum degree . The vertex in has color 4 and its neighbor’s color is 8. In Table 5, among the trees corresponding to the color , the pendant vertex colored 0 is connected first. Later, pendant vertices colored 1 and 2 are connected, respectively. Similarly, some trees are connected to the vertices , , and , .
Theorem 16. There are infinitely many trees containing and with maximum number of holes two.
Proof. The construction of trees is similar to the construction described in Theorem 3. For the construction, we use two-hole irreducible span coloring of given in Figure 2 and Table 3.
Data Availability
The authors confirm that the data supporting the findings of this study are available within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.