On connected degree sequences

On connected degree sequences

Jonathan McLaughlin Department of Mathematics, St. Patrick’s College, Dublin City University, Dublin 9, Ireland jonny$_$mclaughlin@hotmail.com
Abstract

This note gives necessary and sufficient conditions for a sequence of non-negative integers to be the degree sequence of a connected simple graph. This result is implicit in a paper of Hakimi. A new alternative characterisation of these necessary and sufficient conditions is also given.

keywords:
connected graph, degree sequence
Msc:
[2010] 05C40
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poly \xyoptiongraph \xyoptionarc \xyoptionweb \xyoption2cell

1 Introduction

A finite sequence of non-negative integers is called a graphic sequence if it is the degree sequence of some finite simple graph. Erdös and Gallai EG () first found necessary and sufficient conditions for a sequence of non-negative integers to be graphic and these conditions have since been refined by Hakimi Hk () (stated in the sequel as Theorem 3.2) as well as (independently) by Havel Hv (). Alternative characterisations and generalisations are due to Choudum Ch (), Sierksma Hoogeveen SH () and Tripathi et al. TVW10 (), TV03 (), TV07 (). This note states a result which is implicit in Hakimi Hk (), before giving an alternative characterisation of these necessary and sufficient conditions for a finite sequence of non-negative integers to be the degree sequence of a connected simple graph.

2 Preliminaries

Let be a graph where denotes the vertex set of and denotes the edge set of (given that is the set of all -element subsets of ). An edge is denoted in the sequel. A graph is finite when and , where denotes the cardinality of the set . A graph is simple if it contain no loops (i.e. for all or parallel/multiple edges (i.e. is not a multiset). The degree of a vertex in a graph , denoted , is the number of edges in which contain . A path is a graph with vertices in which two vertices, known as the endpoints, have degree and vertices have degree . A graph is connected if there exists at least one path between every pair of vertices in the graph. A tree is a connected graph with vertices and edges. denotes the complete graph on vertices. All basic graph theoretic definitions can be found in standard texts such as BM (), D () or GG (). All graphs in this note are undirected and finite.

3 Degree sequences and graphs

A finite sequence of non-negative integers is called realisable if there exists a finite graph with vertex set such that for all . A sequence which is realisable as a simple graph is called graphic. Given a graph then the degree sequence of , denoted , is the monotonic non-increasing sequence of degrees of the vertices in . This means that every realisable (resp. graphic) sequence is equal to the degree sequence of some graph (resp. simple graph) (subject to possible rearrangement of the terms in ). The maximum degree of a vertex in is denoted and the minimum degree of a vertex in is denoted . In this note all sequences will have positive terms as the only connected graph which has a degree sequence containing a zero is .

The following theorem states necessary and sufficient conditions for a sequence to be realisable (though not necessarily graphic).

Theorem 3.1 (Hakimi)

Given a sequence of positive integers such that for then is realisable if and only if is even and .

To address the issue of when a sequence is graphic, Hakimi describes in Hk () a process he called a reduction cycle and uses it to state the following result.

Theorem 3.2 (Havel, Hakimi)

Given a sequence of positive integers such that for then the sequence is graphic if and only if the sequence is graphic.

4 Degree sequences and connected graphs

Definition 4.1

A finite sequence of positive integers is called connected (resp. connected and graphic) if is realisable as a connected graph (resp. connected simple graph) with vertex set such that has degree for all .

Of course disconnected realisations of connected and graphic degree sequences exist, for example, can be realised as a -cycle or as two disjoint -cycles.

As a graph is connected if and only if it contains a spanning tree, then a simple induction argument on the number of edges shows that every spanning tree of a graph , with , has exactly edges. Hence, a necessary condition for a graph , with , to be connected is that .

The following theorem states necessary and sufficient conditions for a sequence to be connected but not necessarily simple.

Theorem 4.2 (Hakimi)

Given a sequence of positive integers such that for then is connected if and only if is realisable and .

The two main tools used in the proof of Theorem 4.2 are -invariant operations (which leave degree sequences unchanged) and Lemma 4.3 (which appears in Hk () as Lemma ).

Consider a graph with . Given any two edges , where and are all distinct, then is transformed by a -invariant operation into when either

  • and , or

  • and .

Figure 1 shows both -invariant operations.

Figure 1: The two possible -invariant operations on resulting in such that

These -invariant operations are used to prove the following important result.

Lemma 4.3

Let , (with ), be maximally connected subgraphs of such that not all of the are acyclic, then there exists a graph with maximally connected subgraphs such that .

The essence of Lemma 4.3 is presented in Figure 2. Note that worst-case-scenarios are assumed i.e. is a cycle and is acyclic with the graph being disconnected.

Figure 2: has one less connected subgraph than following a -invariant operation

5 Results

The first result, Theorem 5.1, is an explicit statement of a result implicit in Hk (). The second result, Theorem 5.2, is a new alternative characterisation of Theorem 5.1 and has a similar flavour to that of Theorem 3.2.

Theorem 5.1

Given a sequence of positive integers such that for then is connected and graphic if and only if the sequence is graphic and .

Proof.

() Suppose that is connected and graphic. It is required to show that is graphic and that .
As for some simple (connected) graph then for all and there exists a graph with vertex set and edge set such that . As is a simple graph then it follows that is also a simple graph, hence is graphic. As for some (simple) connected graph , where , then as is connected , hence .

() Suppose that is graphic and that . It is required to show that is both graphic and connected.

As is graphic then for all . Adding a term (which is necessarily ) results in also being graphic as in the worst case scenario i.e. where , then which is the degree sequence of the simple graph . Suppose that , then is the degree sequence of a graph with edges and vertices which means that is a tree, hence is connected. If then either for some connected graph or it is possible to apply Lemma 4.3 repeatedly until a graph is found such that is connected and . ∎

The following result is what can be thought of as a connected version of Theorem 3.2. However, note that it is not possible to simply add the word connected to the statement of Theorem 3.2 as is connected but is not connected.

Theorem 5.2

Given a sequence of positive integers such that for then is connected and graphic if and only if the sequence is connected and graphic.

Proof.

() Suppose that is connected. It is required to show that is both graphic and connected.
To show that is graphic it is required to show that is even and that all vertices have degree less than or equal to . Observe that

As is graphic then is even and so is also even. As is graphic then all vertices with must satisfy . All vertices with degree in are necessarily connected to whereas vertices with degree less than may or may not be connected to . It follows that after deleting and all edges containing that the maximum degree which any vertex can have in any is (where ).

To show that is connected it is required to show that i.e. there exists a graph with and such that . As is graphic then .

Let : As then . Not all except in the case where resulting in which is a connected degree sequence. As is connected and then is a leaf of a connected graph and so deleting cannot result in a disconnected graph , hence is connected when .

Let where : As then

Assuming the worst case scenario i.e. , then this gives

which means that

whenever . Hence, is connected when where .

() Suppose that is connected. It is required to show that is both graphic and connected.

As is connected then there exists some with and where the degree of all vertices in is less than or equal to . As (and all are necessarily ) then all vertices in will have degree at most . Observe that

As is graphic then is even and so is also even.

As is connected then there exists some with where as . As then this means that there is at least one edge in which has as an endpoint and some with as the other endpoint.

Figure 3: If is connected and graphic then is connected and graphic

This observation along with the fact that , where , means that is connected. ∎

Example 5.3

An example, using Theorem 5.2, of what Hakimi would term a “set of successive reduction cycles” is shown in Figure 4.

Figure 4: A set of successive reduction cycles

6 Comments

Theorem 3.2 is used in Hk () to check algorithmically when a given sequence is graphic. In a similar manner, Theorem 5.2 suggests an algorithm which can be used to determine if a given sequence is connected and graphic.

References

  • [1] J. A. Bondy and U. S. R. Murty. Graph theory, volume 244 of Graduate Texts in Mathematics. Springer, New York, 2008.
  • [2] S. A. Choudum. A simple proof of the Erdös-Gallai theorem on graph sequences. Bull. Austral. Math. Soc., 33(1):67–70, 1986.
  • [3] R. Diestel. Graph theory, volume 173 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 2000.
  • [4] P. Erdös and T. Gallai. Graphs with prescribed degrees of vertices. Mat. Lapok, 11:264–274 (in Hungarian), 1960.
  • [5] R. Gould. Graph theory. The Benjamin/Cummings Publishing Co. Inc., Menlo Park, CA, 1988.
  • [6] S. L. Hakimi. On realizability of a set of integers as degrees of the vertices of a linear graph. I. J. Soc. Indust. Appl. Math., 10:496–506, 1962.
  • [7] V. Havel. A remark on the existence of finite graphs. Časopis Pěst. Mat., 80:477–480 (in Czech), 1955.
  • [8] G Sierksma and H Hoogeveen. Seven criteria for integer sequences being graphic. J. Graph Theory, 15(2):223–231, 1991.
  • [9] A. Tripathi, S. Venugopalan, and D. B. West. A short constructive proof of the Erdös and €“Gallai characterization of graphic lists. Discrete Math., 310(4):843 – 844, 2010.
  • [10] A. Tripathi and S. Vijay. A note on a theorem of Erdös and Gallai. Discrete Math., 265:417 – 420, 2003.
  • [11] A. Tripathi and S. Vijay. A short proof of a theorem on degree sets of graphs. Discrete Appl. Math., 155(5):670 – 671, 2007.
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