Infinitely Many Trees with Maximum Number of Holes Zero, One, and Two

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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 (2, 1)coloring as a variation of channel assignment problem.The distance between two vertices  and V in a graph , denoted by (, V), is defined as the length of a shortest path between  and V in .An (2, 1)-coloring of a graph  is an assignment : () → {0, 1, 2, . . ., } such that, for every , V in (), |() − (V)| ⩾ 2 if  and V are adjacent and |() − (V)| ⩾ 1 if  and V are at distance 2. The nonnegative integers assigned to the vertices are also called colors.The span of , denoted by  , is max{(V): V ∈ ()}.The span of , denoted by (), is min{ :  is an (2, 1)-coloring of }.An (2, 1)-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 (  ) = 4 for  ⩾ 5; () is either Δ + 1 or Δ + 2 for any tree  with maximum degree Δ.We refer to a tree as Type-I if () = Δ + 1; otherwise it is Type-II.In a graph  with maximum degree Δ, we refer to a vertex V as a major vertex if its degree is Δ; otherwise V 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 () = Δ + 1 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 (2, 1)-span of trees up to twenty vertices.
Fishburn and Roberts [6] have introduced the concept of no-hole (2, 1)-coloring of a graph.If  is an (2, 1)coloring of a graph  with span , then an integer ℎ ∈ {0, 1, 2, . . ., } is called a hole in  if there is no vertex V in  such that (V) = ℎ.An (2, 1)-coloring with no hole is called a no-hole coloring of .Fishburn et al. [7] have introduced the concept of irreducibility of (2, 1)-coloring.An (2, 1)-coloring of a graph  is reducible if there exists another (2, 1)-coloring  of  such that () ⩽ () for all vertices  in  and there exists a vertex V in  such that (V) < (V).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 inhspan of , denoted by  ℎ (), is defined as  ℎ () = min{span  :  is an inh-coloring of }.Fishburn et al. [7] have proved that paths, cycles, and trees are inh-colorable except  3 ,  4 , and stars.In addition to that, they showed that Δ + 1 ⩽  ℎ () ⩽ Δ + 2 where  is any nonstar tree.Laskar et al. [8] have proved that any nonstar tree  is inhcolorable and  ℎ () = ().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 ,   () = 2 if and only if  is a path   ,  > 4. 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 onehole 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.

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.
If {(V 1 ), (V 2 )} = {0, 2}, then 1 and 3 are the holes.If any major vertex V receives a color  other than 0 and 2, then the neighbors of V cannot get the colors 1 and 3 and at least one of  − 1 and  + 1 (if  = Δ + 2,then  − 1).This is not possible as we need Δ + 1 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 {1, 3}, {1, Δ + 1}, or {Δ − 1, Δ + 1}.
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 Δ 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 {1, 3} or {1, Δ+ 1} or {Δ− 1, Δ+ 1}.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 1 and 3 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 (2, 1)-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 0 ⩽  ⩽ Δ + 2 and the color  is not equal to any of the colors  − 1, ,  + 1, 1, and 3 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 twohole 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 1 and Δ + 1 are the holes in .Construction is similar to the previous case using trees in Table 2.
Suppose Δ − 1 and Δ + 1 are the holes in .We use trees in Table 3 for construction.Theorem 4. If T is a tree with   () = 1, then there exist infinitely many trees containing  and with maximum number of holes at least 1.
Proof.Here, we start with a one-hole irreducible (2, 1)span coloring of  having hole ℎ.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,   (  ) ⩾ 1.

Theorem 5.
If  is a tree with   () = 1 and  has no twohole 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  Table 1: Trees connectable to a vertex  colored  in a two-hole tree with 1 and 3 as holes.

Color of vertex Connectable trees Condition
> 3 and all colors greater than 3 less than  adjacent to .

Color of vertex Connectable trees Condition
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   () = 1, then there exist infinitely many trees with maximum number of holes one and containing .

Maximum Number of Holes in Some Type-II Trees
Recall that, in a graph  with maximum degree Δ, we refer a vertex V as a major vertex if its degree is Δ.Otherwise V 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 Δ.
(I)  1 : a tree with an induced  3 consisting of three major vertices.
(II)  2 : a tree with a minor vertex  and at least 3 major vertices adjacent to .
(III)  3 : a tree with a major vertex  and at least Δ−1 major vertices at distance two from , and  2 is not a subtree of the tree.
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  1 ,  2 ,  3 , and  4 .For any tree  with maximum degree Δ, it is clear that   () ⩽ 2. First, we show that   (  ) ⩽ 1,  = 1, 2, 4. Also,   ( 2 ) = 0 if  2 has a vertex adjacent to at least four major vertices.Further, we give a two-hole (2, 1)-irreducible span coloring of  3 if it has exactly Δ − 1 major vertices at distance two from a major vertex and we show that   ( 3 ) ⩽ 1, if  3 has exactly Δ major vertices at distance two from a major vertex.Later, we show that these upper bounds are the exact values by defining (2, 1)-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   ,  = 1, 2, 4,   (  ) ⩽ 1.
Proof.Let V 1 , V 2 , and V 3 be the major vertices of  1 .Since V 1 , V 2 , and V 3 receive three different colors in any (2, 1)coloring, by Lemma 1,  1 cannot have a two-hole irreducible span coloring.Similarly, we can prove that   ( 2 ) ⩽ 1.Now, we consider  4 with labelling as in Figure 1.Suppose that  is a two-hole irreducible span coloring of  4 .Then by Lemma 1, all major vertices of  4 receive colors from {0, 2} or {0, Δ + 2} or {Δ, Δ + 2}.Suppose the major vertices receive 0 and 2. Then 1 and 3 are holes.Without loss and all colors less than  adjacent to  2 0 k  > 3 and all colors greater than 3 less than  adjacent to . of generality, we assume that (V  ) = 0 and (V   ) = 2. Now, one of the pendant vertices adjacent to V  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,   ( 4 ) ⩽ 1.

Theorem 9. If at least four major vertices are adjacent to 𝑤 in
Proof.Recall that  2 is a tree with a vertex  adjacent to at least three major vertices.Let V 1 , V 2 , V 3 , and V 4 be four major vertices adjacent to  in  2 .Suppose that it has a one-hole irreducible (2, 1)-span coloring .Let  1 ,  2 ,  3 , and  4 be the colors received by V 1 , V 2 , V 3 , and V 4 , respectively.Without loss of generality, we assume that 0 ⩽  1 <  2 <  3 <  4 ⩽ Δ + 2. If  1 ̸ = 0, then except  1 − 1 and  1 + 1 all other colors are used to the neighbors of V 1 .Also, except  3 − 1 and  3 + 1, all other colors are used to the neighbors of V 3 .Since  1 − 1,  1 + 1,  3 − 1, and  3 + 1 are four different colors,  cannot have a hole which is a contradiction.So,  1 = 0. Since  is one-hole coloring, the colors  2 − 1,  2 + 1,  3 − 1, and  3 + 1 cannot be four different colors and hence  2 + 1 =  3 − 1 is the hole.Now, a pendant neighbor of V 1 receives  3 which reduces to the hole  3 − 1 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 Table 3: Trees connectable to a vertex  colored  in a two-hole tree with Δ − 1 and Δ + 1 as holes.
> ℎ and all colors greater than ℎ less than  adjacent to .
> ℎ and all colors greater than ℎ less than  adjacent to  and  > ℎ.

Color of vertex Connectable trees Condition
> ℎ and all colors greater than ℎ less than  adjacent to .
T 6  ̸ = ℎ + 1,  > ℎ and all colors greater than ℎ less than  adjacent to . 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 Δ which is also a counterexample for the conjecture.
Figure 3: The tree  3 with exactly Δ major vertices at distance two to . w Figure 6: Irreducible (2, 1)-span coloring of   3 with one hole..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  1 ,   2 ,   2 ,   3 , and  4 and infinitely many two-hole trees containing   3 .
Theorem 13.If the tree  2 with at least five major vertices is a subtree of a tree  with maximum degree same as that of  2 , then   () = 0. ) is 0, first we construct a tree   2 from   2 such that   (  2 ) = 1.We define a one-hole span coloring for   2 as in Figure 8 (  2 is a subtree of   2 ).Since the colors Δ+1 and Δ+2 received by the vertices adjacent to the vertex  are reducible and there is no other color reducible, we connect star  1,Δ−2 to the vertices to make the colors Δ + 1 and Δ + 2 irreducible.The tree obtained is   2 .Now, using Table 5 obtained from Table 4 corresponding to the hole ℎ = Δ and using irreducible one-hole span colorings of  1 ,   2 ,   3 , and  4 given in Theorem 12, we construct infinitely many one-hole trees containing each of the trees  1 ,   2 ,   3 , and  4 , respectively.We get infinitely many trees containing   2 by using irreducible one-hole coloring of   2 given in Figure 8 and using Table 5.
Example 15.In Figure 9, we illustrate the construction of onehole tree as in Theorem 14 for the tree  1 with maximum degree Δ = 7.The vertex  1 in  1 has color 4 and its neighbor's color is 8.In Table 5, among the trees corresponding to the color  = 4, 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  2 ,  3 , and   , 1 ⩽  ⩽ 3.
Theorem 16.There are infinitely many trees containing   3 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   3.

2 , 3
and all colors less than  adjacent to . 0 k 3 <  ⩽ Δ and all colors less than  adjacent to .
> 3 and all colors greater than 3 less than  adjacent to .

3 Figure 9 :
Figure 9: A tree with maximum number of holes one constructed from  1 .

Table 2 :
Trees connectable to a vertex  colored  in a two-hole tree with 1 and Δ + 1 as holes.
k 1 <  ⩽ Δ and all colors less than  adjacent to .

Table 4 :
Trees connectable for a vertex colored  in a one-hole tree with hole ℎ.