Discrete Mathematics 321 (2014) 24–34
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Discrete Mathematics
journal homepage: www.elsevier.com/locate/disc
On the k-residue of disjoint unions of graphs with
applications to k-independence
David Amos a , Randy Davila b , Ryan Pepper c,∗
a
Department of Mathematics, Texas A&M University, United States
b
Computational and Applied Mathematics Department, Rice University, United States
c
Department of Computer and Mathematical Sciences, University of Houston - Downtown, United States
article
info
Article history:
Received 17 January 2013
Received in revised form 10 December 2013
Accepted 12 December 2013
Available online 28 December 2013
Keywords:
Independence
Residue
k-independence
k-residue
Function graphs
Disjoint union lemma
abstract
The k-residue of a graph, introduced by Jelen in a 1999 paper, is a lower bound on the
k-independence number for every positive integer k. This generalized earlier work by
Favaron, Mahéo, and Saclé, by Griggs and Kleitman, and also by Triesch, who all showed that
the independence number of a graph is at least as large as its Havel–Hakimi residue, defined
by Fajtlowicz. We show here that, for every positive integer k, the k-residue of disjoint
unions is at most the sum of the k-residues of the connected components considered
separately, and give applications of this lemma. Our main application is an improvement
on Jelen’s bound for connected graphs which have a maximum degree cut-vertex. We
demonstrate constructively that, in some cases, our extensions give better approximations
to the k-independence number than all known lower bounds—including bounds of Hopkins
and Staton, Caro and Tuza, Favaron, Caro and Hansberg, as well as Jelen’s k-residue bound
itself. Additionally, we apply this disjoint union lemma to prove a theorem for function
graphs (those graphs formed by connecting vertices from a graph and its copy according
to a given function) while simultaneously giving, in this context, different classes of nontrivial examples for which our new results improve on the k-residue, further motivating
our first application of the lemma.
Published by Elsevier B.V.
1. Introduction
Let G = (V , E ) be a simple, finite graph with order n = n(G) and size m = m(G). A set I ⊆ V is an independent set if the
vertices in I are pairwise nonadjacent. The cardinality of a largest independent set in G is called the independence number of G
and is denoted α(G). The problem of computing independence number is known to be NP-hard [14,25]. As such, significant
research has been devoted to finding sharp upper and lower bounds in terms of easily computable invariants (see, e.g.,
[1,11,16,26,32,33]).
A generalization of independent sets was made by Fink and Jacobson [12,13] and later by Hopkins and Staton [22]. Given
a positive integer k, a set I ⊆ V is a k-independent set if the subgraph induced by I has maximum degree at most k − 1. The
k-independence number of a graph G, denoted αk (G), is the cardinality of a largest k-independent set. The computation of
αk was shown to be NP-complete by Jacobson and Peters [23], and a large body of literature exists on k-independence and
related invariants (see [2–5,7,9,10,19,22,24,29]).
The residue of a graph, introduced by Fajtlowicz [8], was conjectured to be a lower bound on the independence number
by his conjecture-making computer program ‘‘Graffiti’’. This conjecture was proven by Favaron, Mahéo, and Saclé [10], and
∗
Corresponding author.
E-mail address: [email protected] (R. Pepper).
0012-365X/$ – see front matter. Published by Elsevier B.V.
http://dx.doi.org/10.1016/j.disc.2013.12.013
D. Amos et al. / Discrete Mathematics 321 (2014) 24–34
25
later by Griggs and Kleitman [17] and Triesch [31]. Jelen generalized the notion of residue to k-residue and showed that
this is a lower bound for k-independence number for every k [24]. In this paper we investigate some properties of k-residue,
extending Jelen’s result and giving improvements under certain conditions. Moreover, we present an infinite family of graphs
for which our result is an improvement on all known tractable lower bounds for k-independence number.
For the entire paper, k is assumed to be a positive integer. The maximum degree of a graph G will be denoted ∆ = ∆(G).
The union of two graphs G and H is the disconnected graph G ∪ H with vertex set V (G) ∪ V (H ) and edge set E (G) ∪ E (H ).
For compactness we denote the union of r copies of G by rG. For any v ∈ V (G), the graph G − {v} is obtained by deleting v
from V (G) and removing any edges incident to v from E (G). We denote the complete graph on n vertices by Kn and the star
on n vertices by Sn (this is equivalent to the complete bipartite graph K1,n−1 ).
Before proceeding, we survey some known results on the k-independence number and introduce a few necessary tools.
2. Known results on k-independence
We mention five tractable lower bounds on the k-independence number, proven by: Hopkins and Staton [22],
Favaron [10], Caro and Tuza [4], Jelen [24], and a corollary to a recent result of Caro and Hansberg [2]. Favaron’s result
was improved upon by Caro and Tuza, while Caro and Tuza’s result was improved upon by Jelen and then later by Caro and
Hansberg. Consequently, we will review only the lower bounds of Hopkins and Staton, Caro and Hansberg, and Jelen—all
three of which are mutually non-comparable, as will be seen in the subsequent examples. We follow the notation used by
Jelen and refer to the bound of Hopkins and Staton as HSk , and the bound of Caro and Hansberg by CHk .
Theorem 1 (Hopkins–Staton [22]). Let G be a graph with maximum degree ∆. Then,
n
αk (G) ≥ HSk (G) :=
1+
 .
∆
k
Theorem 2 (Caro–Hansberg [2]). Let G be a graph with average degree d. Then,1
αk (G) ≥ CHk (G) :=
kn
k + ⌈ d⌉
.
These two bounds are non-comparable. For graphs where the maximum degree is far enough from the average degree,
CHk improves upon HSk . For example, CH1 (Sn ) = 3n and HS1 (Sn ) = 1, thus for n > 3, CH1 > HS1 . On the other hand, for
regular graphs, HSk improves upon CHk . For example, let G be a cubic graph on n vertices and let k = 2. Then HS2 (G) = 2n
and CH2 (G) = 2n
, thus HS2 > CH2 for every n.
5
In order to introduce Jelen’s bound, we need a few concepts. The degree sequence of G, denoted D(G), is the sequence
that lists the degrees of the vertices of G, which we will always write in non-increasing order. When we are required to be
explicit, we will write the degree sequence as the sequence of distinct degrees of G with the number of vertices realizing
each degree in superscript. For example, the degree sequence of the n-vertex star can be written D(Sn ) = {n − 1, 1(n−1) }.
We say a sequence of non-negative integers is graphic if it is the degree sequence of some simple graph. If D is the degree
sequence of G, then G is said to realize D, or, alternatively, G is a realization of D. The next theorem is due to Havel [20] but
is also attributed to Hakimi [18].
Theorem 3 (Havel [20]). A sequence D = {d1 , d2 , . . . , dn } of non-negative integers in non-increasing order is graphic if and only
if d1 ≤ n − 1 and the sequence D′ = {d2 − 1, d3 − 1, . . . , dd1 +1 − 1, dd1 +2 , . . . , dn } is graphic.
The sequence D′ , obtained by deleting d1 from D and reducing the next d1 elements by one, is called the Havel–Hakimi
derivative of D. Higher derivatives will be denoted D(i) . This gives a well known characterization of graphic sequences.
Theorem 4. A sequence D = {d1 , d2 , . . . , dn } of non-negative integers in non-increasing order is graphic if and only if for some
i ∈ {0, 1, . . . , n − 1}, the sequence D(i) consists of n − i zeros.
The process of taking successive Havel–Hakimi derivatives until a sequence of zeros is reached will be called the
Havel–Hakimi process. The residue of a graphic sequence D, denoted R(D), is the
 number
 of zeros in the final sequence of
the Havel–Hakimi process on D. When the context is clear, we will abbreviate R D(G) as simply R(G).
A useful tool in the study of residue is the idea of majorization. Let A = {a1 , a2 , . . . , an } and B = {b1 , b2 , . . . , bn } be two
sequences of order n with the same sum. Then, A is said to majorize B if, for each positive integer t ≤ n, the following is true:
t

i =1
ai ≥
t

bi .
i=1
1 In [2], the authors define a k-independent set as one whose induced subgraph has degree at most k, as opposed to at most k − 1, leading to a slightly
different formulation of the lower bound in the theorem.
26
D. Amos et al. / Discrete Mathematics 321 (2014) 24–34
(a) Before.
(b) After.
Fig. 1. A 2-switch.
If A majorizes B, we write A ≽ B. Favaron et al. proved the following lemma, which was independently discovered by Griggs
and Kleitman.2
Lemma 5 (Favaron, Mahéo and Saclé [11], Griggs and Kleitman [17]). Let D1 and D2 be graphic sequences of the same order for
which D1 ≽ D2 . Then R(D1 ) ≥ R(D2 ).
A third proof of Graffiti’s conjecture was discovered by Triesch in 1996, who introduced the elimination sequence [31].
The elimination sequence of a graphic sequence D, denoted E (D), is the union of the sequence of integers eliminated during
the Havel–Hakimi process and the resulting sequence of zeros.3 In 1999, yet a another proof appeared, due to Jelen [24].
Jelen was able to prove much more
than

 Graffiti’s conjecture by defining a generalization of the residue of a sequence D,
called k-residue and denoted Rk E (D) , based on Triesch’s elimination sequence.
Let D be a graphic sequence and let E = E (D) be the elimination sequence of D. The k-residue of any graph realizing D is
given by the sum
Rk (E ) =
k−1
1
k i=0
(k − i)fi (E ),
(1)
where fi (E ) is the frequency of i in E. Since each graph has
degree sequence, and each degree sequence has a unique
 a unique

elimination sequence, we sometimes write Rk (E ) as Rk D(G) or even Rk (G). Note that if k = 1, then Eq. (1) gives the number
of zeros in E, i.e. R1 (D) = R(D). Jelen’s work now generalizes the results of Favaron et al., Griggs and Kleitman, and Triesch
with the following two theorems.
Lemma 6 (Jelen [24]). Let D1 and D2 be graphic sequences such that D1 ≽ D2 . Then Rk (D1 ) ≥ Rk (D2 ).
Theorem 7 (Jelen [24]). For any graph G, αk (G) ≥ Rk (G).
Jelen shows in [24] that Rk and HSk are non-comparable. In Example 2 of Section 5.1.1 we give a family of graphs for which
Rk > CHk . However, there also exist graphs for which CHk > Rk . For a particular example of the latter, consider any cubic
graph G of order 100. Then R2 (G) = 75
and CH2 (G) = 40. Thus, as claimed, the three bounds are mutually non-comparable.
2
In the concluding remarks of [24], Jelen notes that the Rk bound has room for improvement. In this paper we investigate
the additive behavior of k-residue over the disjoint union of graphs and present ways to improve Theorem 7 for disconnected
graphs and special kinds of connected graphs.
3. Lemmas and tools
The proof of Theorem 3 presented in [27,34] relies on a method of altering the edges of a graph without changing the
degree sequence. Let G be a graph with order n ≥ 4 and suppose u, v, x, y ∈ V (G) are four different vertices such that
uv, xy ∈ E (G) and ux, v y ̸∈ E (G). A 2-switch (illustrated in Fig. 1 and sometimes called a Ryser switch) is performed by
removing the edges uv and xy and adding the edges ux and v y.
These 2-switches can be used to transform a graph G realizing a graphic sequence D into a (possibly non-isomorphic)
graph H with the same degree sequence as G such that H has a maximum degree vertex v adjacent to a set of vertices of
the next ∆(H ) highest degrees (a proof of this is given in the textbooks by Merris [27] and West [34]). Furthermore, graphs
with such a maximum degree vertex have the smallest k-residues among all graphs realizing a given degree sequence, as
we show in the next lemma.
Lemma 8. Let D = {d1 , d2 , . . . , dn } be a graphic sequence with maximum degree ∆ and
derivative D′ . Let G be
 Havel–Hakimi

any graph realizing D and let v ∈ V (G) be a maximum degree vertex. Then Rk (D′ ) ≤ Rk G − {v} .
2 Griggs and Kleitman refer to majorization as dominance. Favaron, Mahéo, and Saclé never name the concept, instead defining the relation A ≥ B for
two sequences A and B of the same length in an equivalent manner to our definition of A ≽ B.
3 The authors acknowledge the potentially ambiguous use of E to represent both the edge set of a graph and the elimination sequence of a degree
sequence. In practice, we believe the meaning will be determined unambiguously by the context.
D. Amos et al. / Discrete Mathematics 321 (2014) 24–34

27

Proof. Clearly the degree sums of D′ and D G − {v} are equal, since both sequences are reached by deleting d1 from D and
′
reducing ∆ terms
by one.

 The reductions in D occurred among the next highest ∆ degrees, while this was not necessarily

the case in D G − {v} . It follows that for any positive integer t ≤ n, the sum of the first t degrees of D G − {v} is at least
′
′
as much as the
 sum of
 the first t degrees of D . Thus D(G − {v}) ≽ D , by definition of majorization. Therefore, by Lemma 6,
Rk (D′ ) ≤ Rk G − {v} . Next we show that for large enough k, Rk is calculated by an explicit formula.
Lemma 9 (Pepper [28]). If k ≥ ∆, then Rk (G) = n −
m
.
k
Proof. Let E be the elimination sequence of D(G). We expand the summand in the definition of k-residue as follows:
k−1
1
Rk (G) =
=
k i =0
k−1

(k − i)fi (E )
fi (E ) −
i=0
k−1
1
k i=1
ifi (E ).
During the Havel–Hakimi process, the entirety of the degree sequence is either eliminated or reduced to zero and, at each
step, the amount eliminated is equal to the amount reduced. Consequently, since the sum of the degree sequence is 2m, the
sum of the elimination sequence is m. We consider two cases: k ≥ ∆ + 1 and k = ∆. For the case when k ≥ ∆ + 1, we have
that,
k−1

k−1

fi (E ) = n and
i =0
ifi (E ) = m.
i=1
Together with the expansion of Rk (G) from above, this implies,
m
Rk (G) = n − .
k
Similarly, when k = ∆,
R∆ (G) =
∆
−1

fi (E ) −
i=0
∆ −1
1 
∆
= (n − f∆ (G)) −
= n−
m
∆
=n−
ifi (E )
i =1
1
∆
m
(m − ∆f∆ (E ))
.
k
Therefore, equality holds in both cases and the theorem is proven.
It is easy to see that the residue of any complete graph is 1, i.e. R(Kn ) = 1. The following lemma generalizes this fact.
Lemma 10. For positive integers k and n,
Rk (Kn ) =

k+1


,
k≤n
2

n − n(n − 1) ,
2k
k ≥ n + 1.
Proof. When k ≥ n + 1, the result follows from Lemma 9, so we focus on the case when k ≤ n. It is easily verified, by
performing the Havel–Hakimi process, that the elimination sequence of Kn is E = {n − 1, n − 2, . . . , 2, 1, 0}, so fi (E ) = 1
for i = 0, 1, . . . , n − 1. Applying the definition of k-residue, we have
k−1
1
(k − i)fi (E )
k i=0

1
=
k + (k − 1) + (k − 2) + · · · + 2 + 1
k
Rk (Kn ) =
=
=
1

k(k + 1)
k

2
k+1
2
. Now we are ready to present the main result of our paper.
28
D. Amos et al. / Discrete Mathematics 321 (2014) 24–34
4. The Disjoint Union Lemma
Few examples are required to discover that residue does not exhibit the same general additivity over the unions of
graphs as other invariants, such as independence number. One such example is the union of two copies of C4 and k = 1.
Then R1 (C4 ∪ C4 ) = 3 but 2R1 (C4 ) = 4. The following theorem is somewhat surprising in this regard. We refer to both this
theorem and its immediate corollary as the Disjoint Union Lemma.
Theorem 11 (The Disjoint Union Lemma). For any two disjoint graphs G and H,
Rk (G ∪ H ) ≤ Rk (G) + Rk (H ).
Proof. We handle the case that k ≥ ∆ first. Let G and H be any two graphs. By Lemma 9, Rk (G ∪ H ) = n(G ∪ H ) −
Since n(G ∪ H ) = n(G) + n(H ) and m(G ∪ H ) = m(G) + m(H ), we have
Rk (G ∪ H ) = n(G) + n(H ) −

= n(G) −
m(G)
m(G) + m(H )
k

k
m(G∪H )
.
k

+ n( H ) −
m(H )

k
= Rk (G) + Rk (H ).
Thus equality holds whenever k ≥ ∆.
For the case when k < ∆, we proceed by induction on n(G ∪ H ). Notice that the result is easily verified for any two graphs
whose orders sum to 3 or less. Assume the theorem is true for all disjoint graphs whose orders sum to less than p, where
p ≥ 4.
Let G and H be disjoint graphs whose orders sum to p. We may assume without loss of generality that both G and H
have at least two vertices since the result is easily verified otherwise. Moreover, we may assume that there is a vertex v of
maximum degree in G ∪ H which is in G (if not, relabel the graphs).
Now, perform 2-switches to G, if needed, until v is adjacent to a set of vertices realizing the degrees d2 , d3 , . . . , d∆+1 of
D(G). Let G′ = G − {v}. Note that D(G′ ) = D′ (G), and both sequences have order p − 1. Now, the elimination sequence
of G and G′ are identical except that f∆ (E (G′ )) = f∆ (E (G)) − 1. However,
since k < ∆, the f∆ term is not
in the

 included


k-residue formula for either E (G) or E (G′ ). Thus Rk (G′ ) = Rk D′ (G) = Rk (G). Furthermore, Rk D′ (G ∪ H ) ≤ Rk D(G′ ∪ H ) ,
by Lemma 8.
Finally, applying the inductive hypothesis and the facts discussed above, we have the following chain of inequalities
whenever k < ∆,
Rk (G ∪ H ) = Rk D′ (G ∪ H )




≤ Rk D(G′ ∪ H )


≤ Rk D(G′ ) + Rk (H )
≤ Rk (G) + Rk (H ). Theorem 11 is easily extended to the number of components of any disconnected graph.
Corollary 12. For any disconnected graph G with p components Gi ,
Rk (G) ≤
p

Rk (Gi ).
i=1
Equality for Theorem 11 is satisfied whenever k ≥ ∆, as was shown in the proof of the theorem. However, if the
frequencies of the degrees in the elimination sequence of the union of two graphs is equal to the sum of the frequencies in
the elimination sequences of the two graphs, considered individually, then the definition of k-residue guarantees that the
k-residue of the union is equal to the sum of the k-residues of the individual graphs, for every k.
We state this more compactly in the following observation. By the union of two elimination sequences, we mean the
sequence obtained by simply combining the two sequences and arranging the resulting sequence in non-increasing order.
Observation 13. For two graphs G and H, if E (G ∪ H ) = E (G) ∪ E (H ) then Rk (G ∪ H ) = Rk (G) + Rk (H ) for every k.
Remark. Note that the converse of Observation 13 is not true. For example, consider the sequence D = {32 , 26 , 12 }, which
has the disjoint realization of two copies of a graph H obtained by adding a degree one vertex to any vertex of a C4 . It is easily
verified that R1 (D) = 2R1 (H ) = 4. However, E (D) = {31 , 22 , 13 , 04 }, while E (H ) = {31 , 12 , 02 }. That is, R1 (D) = 2R1 (H )
but E (D) ̸= E (H ) ∪ E (H ).
D. Amos et al. / Discrete Mathematics 321 (2014) 24–34
29
Fig. 2. S5 ∪ K̃4 .
The union of two complete graphs realizes Observation 13, as can be seen by performing the Havel–Hakimi process. In
light of Corollary 12, this can be extended to the following result, which shows that the Disjoint Union Lemma is satisfied with
equality for degree sequences realizable by the union of complete graphs—with the k-residue function behaving analogously
to a linear operator under these circumstances.
Theorem 14. Let n1 , n2 , . . . , np , c1 , c2 , . . . , cp be a positive integers. Then, for every k,

Rk
p


ci D(Kni )
=
i=1
p

ci Rk D(Kni ) . 

i =1
Theorem 14 will prove fruitful in simplifying calculations in the examples to follow.
In the next section, we show that families of graphs exist for which the difference between the sum of the residues of
the components of a disjoint union and the k-residue of the union can be arbitrarily large. These are precisely the graphs for
which the Disjoint Union Lemma will offer the greatest improvement over Theorem 7 as a lower bound for αk .
5. Applications of the Disjoint Union Lemma
We know that Rk (G) + Rk (H ) ≤ αk (G) + αk (H ) by Jelen’s result in Theorem 7. Moreover, since k-independence number
is additive componentwise, we have αk (G ∪ H ) = αk (G) + αk (H ). These facts, combined with The Disjoint Union Lemma,
justify the next result.
Theorem 15. For every positive integer k and any disconnected graph G with p components Gi ,
Rk (G) ≤
p

Rk (Gi ) ≤
i=1
p

αk (Gi ) = αk (G). i=1
Remark. Recall that the Disjoint Union Lemma is sharp for unions of complete graphs. The result is more interesting, with
regards to Theorem 15, when the difference between the two sides of the inequality in the Disjoint Union Lemma is large.
We will construct a family of graphs, each graph a disjoint union of two graphs, such that the sum of the k-residues of the
components grows arbitrarily larger than the k-residue of the union, while staying relatively close to the k-independence
number. Indeed, for the graphs in our family, the difference between the k-independence number and the sum of the kresidues is a function of k only.
Example 1. Let r and k be positive integers. Consider the disjoint union of the star on r + 1 vertices and the corona of the
complete graph on r vertices, denoted K̃r , and obtained by attaching a degree one vertex to each vertex of Kr (see Fig. 2).
First we compute Rk (Sr +1 ). Since D(Sr +1 ) = {r , 1r }, then D′ (Sr +1 ) = {0r } and E (Sr +1 ) = {r , 0r }. We only consider the
case that k ≤ r, since otherwise equality holds for Theorem 11. Thus, whenever k ≤ r
Rk (Sr +1 ) =
k−1
1
1
k i =0
k
(k − i)fi (E (Sr +1 )) =
(kr ) = r .
To compute Rk (K̃r ) we note that D(K̃r ) = {r r , 1r }. The Havel–Hakimi process is as follows:
D′ (K̃r ) = {(r − 1)(r −1) , 1(r −1) , 0}
D′′ (K̃r ) = {(r − 2)(r −2) , 1(r −2) , 02 }
D′′′ (K̃r ) = {(r − 3)(r −3) , 1(r −3) , 03 }
..
.
D(r ) (K̃r ) = {0r }.
(2)
30
D. Amos et al. / Discrete Mathematics 321 (2014) 24–34
Hence E (K̃r ) = {r , r − 1, r − 2, . . . , 2, 1, 0r }. Therefore, whenever k ≤ r
Rk (K̃r ) =
k−1
1
(k − i)fi (E (K̃r ))
k i =0

1
kr + (k − 1) + (k − 2) + · · · + 2 + 1
=
k
=
1

kr +
k
=r+
k(k − 1)
k−1
2

2
.
(3)
Finally, we compute Rk (Sr +1 ∪ K̃r ). Observe that
D(Sr +1 ∪ K̃r ) = {r (r +1) , 12r } = D(Kr +1 ∪ rK2 ).
By Theorem 14,
Rk (Sr +1 ∪ K̃r ) = Rk (Kr +1 ∪ rK2 ) = Rk (Kr +1 ) + rRk (K2 ).
Since ∆(K2 ) = 1 and 1 ≤ k ≤ r, Lemmas 9 and 10 give
k+1
Rk (Kr +1 ) + rRk (K2 ) =
2

+ 2−
1
k

r.
(4)
Therefore, using Eqs. (2)–(4),
Rk (Sr +1 ) + Rk (K̃r ) − Rk (Sr +1 ∪ K̃r ) = 2r +
= 2r +
=
r
k
k−1
2
k−1
2

−
2−
− 2r +
r
k
1

k
−
r+
k+1

2
k+1
2
−1
which grows arbitrarily large as r → ∞.
We now show that αk (Sr +1 ∪ K̃r ) − (Rk (Sr +1 ) + Rk (K̃r )) can be quite small and, in fact, depends only on k. The kindependence number of Sr +1 is easily computed.
αk (Sr +1 ) =

r,
r + 1,
1≤k≤r
k ≥ r + 1.
(5)
For K̃r , if k ≥ r + 1 then αk (K̃r ) = 2r. If 1 ≤ k ≤ r, observe that taking each of the r degree one vertices and k − 1 vertices
of degree r produces a largest k-independent set. Hence,
αk (K̃r ) =

r + k − 1,
2r
1≤k≤r
k ≥ r + 1.
(6)
As before, we only consider the case that 1 ≤ k ≤ r, since equality for the Disjoint Union Lemma is satisfied otherwise.
Applying Eqs. (2), (3), (5) and (6), observe now that the difference,

αk (Sr +1 ∪ K̃r ) − Rk (Sr +1 ) + Rk (K̃r ) = (2r + k − 1) − 2r +


k−1
2

=
k−1
2
is a function of k only. Note especially that, when k = 1, the sum of the residues of the components is simultaneously equal
to the independence number, and grows arbitrarily larger than the residue of the union as r → ∞. 5.1. The Disjoint Union Lemma and connected graphs
In this section, we use the Disjoint Union Lemma to prove, for graphs with a maximum degree cut vertex, a theorem
analogous to Theorem 11. Then we prove an upper bound for the k-residue for a type of connected graph called a function
graph.
D. Amos et al. / Discrete Mathematics 321 (2014) 24–34
31
Fig. 3. The result of Theorem 16 does not hold for c.
5.1.1. Graphs with a maximum degree cut vertex
Theorem 16. Let G be a graph with a maximum degree cut vertex c and let G1 , G2 , . . . , Gp be the components of G − {c }. Then
for k ≤ ∆(G),
Rk (G) ≤
p

Rk (Gi ) ≤
p

i=1
αk (Gi ) ≤ αk (G).
i=1
Proof. First we show that Rk (G) ≤
p
Rk (Gi ). Since k ≤ ∆(G), Rk (G) = Rk D′ (G) . It follows from Lemma 8 that,

i=1

Rk (G) = Rk D′ (G) ≤ Rk G − {c } .




p
Since D G − {c } = D(∪i=1 Gi ), the Disjoint Union Lemma yields,




p
p



Rk (Gi ).
Rk (G) ≤ Rk G − {c } = Rk
Gi ≤

i=1
i =1
The second inequality in the theorem follows from Theorem 7. To see that the last inequality
p holds, let I be a largest
k-independent set of G − {c }. Since I is also a k-independent set in G, it follows that αk (G) ≥
i=1 αk (Gi ).
Remark. To obtain the best lower bound for k-independence number from Theorem 16, one should look for a maximum
degree cut vertex that maximizes the sum of the k-residues of the components of G − {c }. Additionally, we note that the
conclusion of Theorem 16 does not hold for every cut vertex. For example, consider the graph in Fig. 3, we will call it H.
Removing c produces two copies of K3 . The elimination sequences of H and 2K3 are easily computed: E (H ) = {3, 22 , 1, 03 }
and E (2K3 ) = {22 , 12 , 02 }. Note that R1 (H ) ̸= R1 (2K3 ).
We now present a family of graphs for which Theorem 16 gives a significantly better lower bound for k-independence
number than Theorem 7.
Example 2. Let k, p, and r be positive integers such that p ≥ 2 and k ≤ r. Let Gp,r be a graph obtained by taking p copies of
K̃r (defined in Example 1) and joining the degree one vertices of each K̃r to a new vertex v (see Fig. 4). Note that v is a cut
vertex and that deg(v) = pr = ∆(Gp,r ). Hence Rk (Gp,r ) ≤ pRk (K̃r ) by Theorem 16 above.
Suppose p = 3t for some positive integer t and 2 ≤ k < r. Observe that,
D(Gp,r ) = {pr , r pr , 2pr }
D′ (Gp,r ) = {(r − 1)pr , 2pr }.
In D′ (Gp,r ), the term (r − 1)pr corresponds to the degree sequence of p copies of Kr . Furthermore, since p = 3t for some
positive integer t, the term 2pr = 23tr corresponds to the degree sequence of tr copies of K3 . That is,
D′ (Gp,r ) = D(pKr ∪ trK3 ).
Since k < r , Rk (Gp,r ) = Rk (D′ (Gp,r )), applying Lemma 10, Theorem 14 and the equation above, we get,
Rk (Gp,r ) = Rk (pKr ∪ trK3 )
= pRk (Kr ) + trRk (K3 )




3
k+1
+ tr 3 −
=p
2

=p r+
k
k+1
2
−
r
k

.
32
D. Amos et al. / Discrete Mathematics 321 (2014) 24–34
Fig. 4. G3,4 .
Furthermore, pRk (K̃r ) = p(r + k−2 1 ), as was shown in Example 1. Thus we have,

pRk (K̃r ) − Rk (Gp,r ) = p r +
k−1


−p r +
2
k+1
2
−
r

k

=p
r
k

−1
which grows arbitrarily large as p → ∞ or r → ∞. Thus the sum of the k-residues of the components of Gp,r − {v} can be
arbitrarily larger than the k-residue of Gp,r , giving an improvement on Theorem 7. In the next section, we show that for this
family, Theorem 16 is a much better lower bound for k-independence than both HSk and CHk . 5.1.2. Comparison of Theorem 16 to CHk and HSk
To see that the Disjoint Union Lemma can, in some cases, yield better approximations than the lower bounds mentioned
in the Introduction, consider the family Gp,r defined in Example 2. Note that, in terms of p and r, we have n(Gp,r ) =
pr (r +3)
pr (r +3)
2pr + 1, m(Gp,r ) =
and ∆(Gp,r ) = pr. Hence the average degree is d = 2pr +1 .
2
Let c be the maximum degree cut vertex of Gp,r and let G1 , G2 , . . . , Gj be the components of Gp,r − {c }. Recall from
j
Theorem 16 that Rk (Gp,r ) ≤
i=1 Rk (Gi ) ≤ αk (Gp,r ). Moreover, taking a largest k-independent set in each copy of K̂r
produces a largest independent set in Gp,r (since k < r). Thus, from the results in Example 2,
αk (Gp,r ) = p(r + k − 1) and
p


Rk (Gi ) = p r +
k−1
2
i=1

.
Choose p = r = k2 and recall that p is a multiple of 3. We may express αk (Gp,r ),
HSk (Gp,r ) as follows:
3
αk (Gp,r ) = p(r + k − 1) = p2 + p 2 − p


3
j

k−1
p2
p
Rk (Gi ) = p r +
= p2 +
−
2
i=1

Rk (Gp,r ) = p
k+1
2

+
2
pr
3

3−
3
k
2
3

= p2 −
p2
2
+
p
2
j
i =1
Rk (Gi ), Rk (Gp,r ), CHk (Gp,r ) and
D. Amos et al. / Discrete Mathematics 321 (2014) 24–34
CHk (Gp,r ) =
kn
k + ⌈ d⌉
p(2p2 + 1)
=

1
p2 +
HSk (Gp,r ) =
∆ =
k
3
 ∼ 4p 2
2p2 + 1
n
1+
pr (r +3)
2pr +1
1
 ∼ 2p 2 .

1+
33
p2
1
p2
Here we use ∼ to mean ‘‘asymptotically equal to’’. Observe that αk (Gp,r ) ∼
can be arbitrarily larger than any of the other three bounds.
j
i=1
Rk (Gi ) ∼ Rk (Gp,r ). However,
p
i =1
Rk (Gi )
5.1.3. Function graphs
We define a function graph, denoted (G, f ), as follows. Let G1 and G2 be two copies of some connected graph G. Then (G, f )
is formed by adding edges to G1 ∪ G2 according to a function f : V (G1 ) → V (G2 ).
Function graphs were introduced and studied by Stephen Hedetniemi in [21]. More recently, the colorability and planarity
of function graphs have been studied by Chen et al. in [6]; independence in function graphs is studied by Gera et al. in [15].
We will use the Disjoint Union Lemma to prove a bound on the k-residue of a function graph (G, f ). Incidentally, the graphs
(G, f ∗ ), described in the proof below, give more non-trivial instances of graphs with a maximum degree cut vertex and for
which Theorem 16 can be used to yield improvements over Jelen’s k-residue bound for k-independence.
Theorem 17. Let (G, f ) be a function graph. Then Rk (G, f ) ≤ 2Rk (G) whenever k ≤ ∆(G).
Proof. Let u ∈ V (G2 ) be a maximum degree vertex and let f ∗ be the constant function which maps each v ∈ V (G1 ) to u.
First we will show that for any function f : V (G1 ) → V (G2 ), the following
valid: Rk (G, f ) ≤ Rk (G, f ∗ ). Let
ninequality
is
n
∗
D(G, f ) = {d1 , d2 , . . . , dn } and D(G, f ∗ ) = {d∗1 , d∗2 , . . . , d∗n }. Observe that
d
=
(G, f ∗ ) all n edges
i =1 i
i=1 di . Since, in
t
t
∗
∗
added by f are incident to u, which is not necessarily the case in (G, f ), observe also that for every t ≤ n, i=1 di ≤
i=1 di .
∗
∗
It follows that D(G, f ) ≽ D(G, f ), by
definition
of
majorization,
and
hence
R
(
G
,
f
)
≥
R
(
G
,
f
)
,
by
Lemma
6.
k
k

Now, Rk (G, f ∗ ) = Rk D′ (G, f ∗ ) , since k ≤ ∆(G). Perform 2-switches to G2 , if needed, until u is adjacent to the next
∗
∗
∆(G) vertices of highest degree of G2 . Delete u from (G, f ) and observe
 that (G, f ) − {u}is the disjoint
 union of G1 and
G′ , where G′ is a realization of D′ (G2 ). By Lemma 8, Rk D′ (G, f ∗ ) ≤ Rk (G, f ∗ ) − {u} = Rk D(G1 ∪ G′ ) . From the Disjoint
Union Lemma, Rk D(G1 ∪ G′ ) ≤ Rk (G1 ) + Rk (G′ ). Now, Rk (G1 ) = Rk (G) and Rk (G′ ) = Rk (G), since G1 is just a copy of G and
k ≤ ∆(G). It follows that Rk (G1 ) + Rk (G′ ) = 2Rk (G). Putting this all together, we have,


Rk (G, f ) ≤ Rk (G, f ∗ ) ≤ Rk (G, f ∗ ) − {u} = Rk D(G1 ∪ G′ ) ≤ 2Rk (G). 



6. Concluding remarks
We have seen how the Disjoint Union Lemma can give improvements on Theorem 7 for both disjoint unions and
connected graphs with a maximum degree cut vertex. The examples given in this paper rely heavily on the result of
Theorem 14; that is, that equality for the Disjoint Union Lemma is satisfied by degree sequences realizable by a disjoint
union of complete graphs. This fact follows from Observation 13, which says that, for the union of two graphs A and B,
equality holds for the Disjoint Union Lemma whenever E (A ∪ B) = E (A) ∪ E (A).
However, other kinds of degree sequences exhibit this property, not just those from unions of complete graphs. For
example, consider the degree sequence {28 }, which has the disconnected realization C3 ∪ C5 . A few simple calculations
verify that E (C3 ∪ C5 ) = {23 , 12 , 03 }, E (C3 ) = {2, 1, 0} and E (C5 ) = {22 , 1, 02 }. That is, E (C3 ∪ C5 ) = E (C3 ) ∪ E (C5 ), but the
degree sequence D(C3 ∪ C5 ) = {28 } is not realizable by the union of complete graphs. Notice that this is the same degree
sequence as the example given in the first paragraph of Section 4, and that R1 (D) = R1 (C3 ) + R1 (C5 ). One realization of this
sequence gives equality for the Disjoint Union Lemma, while another does not.
There are two problems, then, left unanswered. These are: (1) characterize the degree sequences with disconnected
realizations that have the property described in Observation 13 and, (2) more generally, characterize equality for the Disjoint
Union Lemma. The example given in the remark after Observation 13 shows that these two problems, although related, are
not equivalent to each other.
Finally, we would like to mention that Lemmas 5 and 6 actually show that residue and k-residue are examples of a
special kind of function from the theory of inequalities [30]. Given two sequences of real numbers A and B with order n,
and a function f that maps those sequences to a real number, f is a Schur convex function provided that A ≽ B implies that
f (A) ≥ f (B). Hence, both residue and k-residue are Schur convex functions—a fact that seems worth exploring further.
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