A Note on the Warmth of Random Graphs with Given Expected Degrees
We consider the random graph model for a given expected degree sequence . Warmth, introduced by Brightwell and Winkler in the context of combinatorial statistical mechanics, is a graph parameter related to lower bounds of chromatic number. We present new upper and lower bounds on warmth of . In particular, the minimum expected degree turns out to be an upper bound of warmth when it tends to infinity and the maximum expected degree with .
Let be a graph with vertex set and edge set . For graphs and , a function is said to be a graph homomorphism  if it induces a map between edges . Denote by the set of all homomorphisms of a graph to a graph . Let denote the -branching rooted tree (with the root having degree ); see Figure 1 for an illustration. A map in is said to be cold if there is a vertex of such that for any no agrees with on the vertices at distance from the root but has . We say that is -warm if does not contain any cold maps. Moreover, the warmth, , of is defined to be the largest for which is -warm. By definition, for any finite and connected graph , and if and only if is bipartite.
Warmth is a graph parameter introduced by Brightwell and Winkler  in the context of combinatorial statistical physics. It is closely related to the chromatic number of a graph, which is the smallest positive integer that is not a root of the chromatic polynomial (see, e.g., ). It was shown that [2, Theorem 5.1] for any unlooped graph the warmth of is at most its chromatic number. A natural question to ask would be what the warmth of a graph looks like in a typical graph or random graphs . Recently, Fadnavis and Kahle  established some upper and lower bounds for Erdös-Rényi random graphs as well as random regular graphs. The main finding is that warmth is often much smaller than chromatic number for random graphs. We mention that most of the parameters examined in random graph theory are monotone with respect to the addition (or deletion) of edges [4, 6]. However, warmth is not such a parameter, which makes it difficult to study in random graph settings.
In this paper, motivated by the work of , we study the upper and lower bounds of warmth in a general random graph model . For a given sequence , is defined as follows. Each potential edge between vertices and is chosen with probability and is independent of other edges, where Here, we assume that and define . An immediate consequence of (1) is that the expected degree at a vertex is exactly . Hence, is the expected average degree.
This model, known as the Chung-Lu model, was first proposed in . The classical Erdös-Rényi random graph can be viewed as a special case of by taking expected degree sequence . Many graph properties, such as component structure [8, 9], average distance , hyperbolicity , and spectral gap [12–14], have been explored for this model. We refer the reader to monograph  for detailed backgrounds and varied related results.
2. Main Results
In this section we establish our main results for upper and lower bounds. We say an event holds asymptotically almost surely (a.a.s.), if it holds with probability tending to 1 as . The asymptotics , , , and are used in their standard sense . For example, let and be two sequences of positive real numbers. Consider means ; consider means that there exists some constant such that for all large enough ; consider means that there exists some constant such that for all large enough ; consider means that there exists some constant such that for all large enough .
For a sparse random graph , we may upper bound its warmth using minimum expected degree.
Theorem 1 (upper bound). For a random graph , suppose the maximum expected degree with and the minimum expected degree as . Then, for , one has
The authors in [5, Theorem 3.1] showed that, for sparse Erdös-Rényi random graph with for some , a.a.s. On the other hand, it is well known that [4, 16] the chromatic number in this regime tends to infinity a.a.s. Therefore, the warmth of is much smaller than its chromatic number.
Our Theorem 1, nevertheless, provides an example where warmth may be close to chromatic number. To see this, we choose , , and . The main result in  then concludes that (in a slightly different formulation, where precise degrees rather than expected degrees are specified) the chromatic number a.a.s. On the other hand, Theorem 1 yields a comparable upper bound .
For dense random graphs, we have the following lower bound.
Theorem 2 (lower bound). For a random graph , suppose for any . Then, for and , one has
We remark that the above result implies Theorem 3.4 in  when and . In view of [2, Theorem 5.1] (as mentioned in Section 1), Theorem 2 provides an alternative approach to obtaining lower bounds for .
For a graph , let denote the minimum degree of . For a vertex , the neighborhood of is denoted by , and, for a subset , the neighborhood of is defined as . A collection of subsets of is called a -stable family if for any there are such that .
We will need the following lemma to prove Theorem 1.
Lemma 3 (see ). Given a graph and a natural number , is not -warm if and only if there is a -stable family of subsets of .
Proof of Theorem 1. Since , for , we have as using a concentration inequality . Set . We will prove that .
Now consider consisting of all singleton vertices of . A vertex set is called an -representative  of if such that all of them are not in the neighborhood for any vertex . Therefore, by Lemma 3, it suffices to prove that every vertex has an -representative.
Suppose . Let denote the event that are in the neighborhood of . Hence,
Recall that as shown in the beginning of the proof. Let and be some disjoint subsets of the neighbors of with for . For and , denote by the event that for some . Therefore, the disjointness and inequality (4) imply Thus, the probability that some vertex does not have an -representative is bounded from above by which tends to zero since with and . This completes the proof of Theorem 1.
Proof of Theorem 2. Set . By definition, we need to prove that contains no cold maps.
In what follows, we proceed with the similar lines of reasoning of Section 5 in . We label the vertices of -branching rooted tree according to Figure 1 with the root labeled 0 and its children , sequently.
Let denote the truncated version of -branching rooted tree with vertices, labeled to . We suppose and has vertices up to level and vertices from level of , where . Hence, we have and . The number of leaves of can be calculated as The leaves are labeled as . Denote by the set of roots and leaves; that is, .
For a graph and a function , is said to be -extendable if there is a homomorphism such that . Hence, if every function is -extendable, then contains no cold maps, and thus the proof will be complete.
Now let which is bounded away from for large enough . We claim the following.
Claim 1. For and , the probability that a function with and is not -extendable is at most for some constants .
Proof of Claim 1. We use Janson’s inequality  to prove Claim 1. Note that by definition (8) the assumptions for any are equivalent to
Let . Then, from (9) and (10), the probability that there does not exist such a for a particular choice of is at most and at least .
Let We have . For , denote by the set of vertices of ordered according to its labeling. Denote by the event that governed by is a homomorphism. Hence, the above discussion implies where denotes the complement of the event .
Let be a subgraph of induced by . Let mean that the edges of and have a nontrivial intersection. We apply Janson’s inequality  and (12) to get where and .
From (9) and (12) we have To estimate , we denote by the maximum over all pairs of the number of edges that and intersect in vertices. Since the edge sets of and have a nontrivial intersection, we get . Hence by (10) since . Noting that there are such pairs of and , we obtain
We choose such that and . Hence, estimates (14) and (16) readily yield
Note that by our choice of . Thus, inequality (13) in conjunction with estimates (14) and (17) means that for some constant , which then concludes the proof of the claim.
Since there are at most choices for , the probability that a homomorphism (with ) does not exist for at least one choice is at most , which tends to zero as . Consequently, with probability one, every map is -extendable for some . The proof of Theorem 2 is then complete.
In this paper we presented upper and lower bounds for the warmth of random graphs with given expected degrees. Our results indicate that the warmth of a typical dense graph is smaller than (but can be rather close to) its chromatic number, shedding some insight on the universal upper bound . It is worth noting that Fadnavis and Kahle  showed that a typical sparse graph has much smaller warmth than chromatic number.
We mention that the degree distributions of the random graph models studied in this paper and  are more or less homogeneous (namely, Poisson-like). It would be interesting to know the behavior of warmth for heterogeneously connected graphs or digraphs [18–20], which are ubiquitous in real-world systems and investigate further the influence of maximum/minimum degrees on the warmth as hinted in Theorem 1.
Conflict of Interests
The author declares that there is no conflict of interests.
The author expresses his sincere gratitude to the anonymous referees and the editor for the careful reading of the original paper and the useful comments that helped to improve the presentation of results.
B. Bollobas, Random Graphs, Cambridge University Press, Cambridge, UK, 2001.View at: MathSciNet
F. Chung and L. Lu, Complex Graphs and Networks, American Mathematical Society, Boston, Mass, USA, 2006.
A. Coja-Oghlan and A. Lanka, “The spectral gap of random graphs with given expected degrees,” The Electronic Journal of Combinatorics, vol. 16, article R138, 2009.View at: Google Scholar