A Note on the Warmth of Random Graphs with Given Expected Degrees

Abstract

We consider the random graph model 𝐺(𝐰) for a given expected degree sequence 𝐰=(𝑤₁,𝑤₂,…,𝑤_𝑛). 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 𝐺(𝐰). 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 𝑚=𝑂(𝑛^𝛼) with 0<𝛼<1/2.

1. Introduction

Let 𝐺=(𝑉(𝐺),𝐸(𝐺)) be a graph with vertex set 𝑉(𝐺) and edge set 𝐸(𝐺). For graphs 𝐻 and 𝐺, a function 𝜙∶𝑉(𝐻)→𝑉(𝐺) is said to be a graph homomorphism [1] if it induces a map between edges 𝐸(𝐻)→𝐸(𝐺). Denote by Hom(𝐻,𝐺) the set of all homomorphisms of a graph 𝐻 to a graph 𝐺. Let 𝑇^𝑑 denote the 𝑑-branching rooted tree (with the root having degree 𝑑); see Figure 1 for an illustration. A map 𝜙 in Hom(𝑇^𝑑,𝐺) is said to be cold if there is a vertex 𝑣 of 𝐺 such that for any 𝑘 no 𝜓∈Hom(𝑇^𝑑,𝐺) agrees with 𝜙 on the vertices at distance 𝑘 from the root 𝑟 but has 𝜓(𝑟)=𝑣. We say that 𝐺 is 𝑑-warm if Hom(𝑇^𝑑−2,𝐺) does not contain any cold maps. Moreover, the warmth, warmth(𝐺), of 𝐺  is defined to be the largest 𝑑 for which 𝐺 is 𝑑-warm. By definition, for any finite and connected graph 𝐺, warmth(𝐺)≥2 and warmth(𝐺)=2 if and only if 𝐺  is bipartite.

Figure 1 

The labeling of vertices for the 3-branching rooted tree.

Warmth is a graph parameter introduced by Brightwell and Winkler [2] 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., [3]). It was shown that [2, Theorem 5.1] for any unlooped graph 𝐺  the warmth of 𝐺 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  [4]. Recently, Fadnavis and Kahle  [5] 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 [5], we study the upper and lower bounds of warmth in a general random graph model 𝐺(𝐰). For a given sequence 𝐰=(𝑤₁,𝑤₂,…,𝑤_𝑛), 𝐺(𝐰) is defined as follows. Each potential edge between vertices 𝑣_𝑖 and 𝑣_𝑗 is chosen with probability 𝑝_𝑖𝑗 and is independent of other edges, where 𝑝_𝑖𝑗=𝑤_𝑖𝑤_𝑗∑^𝑛_𝑘=1𝑤_𝑘. (1) Here, we assume that 𝑤²_max=max_𝑖𝑤²_𝑖<∑^𝑛_𝑘=1𝑤_𝑘 and define 𝑛𝑑=∑^𝑛_𝑘=1𝑤_𝑘. An immediate consequence of (1) is that the expected degree at a vertex 𝑣_𝑖 is exactly 𝑤_𝑖  [7]. Hence, 𝑑  is the expected average degree.

This model, known as the Chung-Lu model, was first proposed in [8]. The classical Erdös-Rényi random graph 𝐺(𝑛,𝑝) can be viewed as a special case of 𝐺(𝐰) by taking expected degree sequence 𝐰=(𝑝𝑛,𝑝𝑛,…,𝑝𝑛). Many graph properties, such as component structure [8, 9], average distance [10], hyperbolicity [11], and spectral gap [12–14], have been explored for this model. We refer the reader to monograph [7] for detailed backgrounds and varied related results.

The rest of the note is organized as follows. We state and discuss the upper and lower bounds for warmth in Section 2. Section 3 contains the proofs. A brief conclusion is drawn in Section 4.

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 𝑛→∞. The asymptotics 𝑜, 𝑂, Ω, and Θ are used in their standard sense [15]. For example, let 𝑓(𝑛) and 𝑔(𝑛) be two sequences of positive real numbers. Consider 𝑓(𝑛)=𝑜(𝑔(𝑛)) means lim_𝑛→∞𝑓(𝑛)/𝑔(𝑛)=0; consider 𝑓(𝑛)=𝑂(𝑔(𝑛)) means that there exists some constant 𝐶>0 such that 𝑓(𝑛)≤𝐶𝑔(𝑛) for all large enough 𝑛; consider 𝑓(𝑛)=Ω(𝑔(𝑛)) means that there exists some constant 𝐶>0 such that 𝑓(𝑛)≥𝐶𝑔(𝑛) for all large enough 𝑛; consider 𝑓(𝑛)=Θ(𝑔(𝑛)) means that there exists some constant 𝐶>0 such that 𝐶⁻¹≤𝑓(𝑛)/𝑔(𝑛)≤𝐶 for all large enough 𝑛.

For a sparse random graph 𝐺(𝐰), we may upper bound its warmth using minimum expected degree.

Theorem 1 (upper bound). For a random graph 𝐺∈𝐺(𝐰), suppose the maximum expected degree 𝑤_max=𝑚=𝑂(𝑛^𝛼) with 0<𝛼<1/2 and the minimum expected degree 𝑤_min=𝜔(𝑛)→∞ as 𝑛→∞. Then, for 0<𝛿<1, one has 𝑤𝑎𝑟𝑚𝑡ℎ(𝐺)≤(1−𝛿)𝜔(𝑛)+1𝑎.𝑎.𝑠. (2)

The authors in [5, Theorem 3.1] showed that, for sparse Erdös-Rényi random graph 𝐺(𝑛,𝑝) with 𝑝=𝑂(𝑛^−𝛼) for some 𝛼>0, warmth(𝐺(𝑛,𝑝))≤⌊1/𝛼+2⌋ a.a.s. On the other hand, it is well known that [4, 16] the chromatic number 𝜒(𝐺(𝑛,𝑝)) in this regime tends to infinity a.a.s. Therefore, the warmth of 𝐺(𝑛,𝑝) 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 𝑤_min=𝑛^𝛼/ln𝑛, 𝑑=Θ(𝑛^𝛼), and 𝛼<1/4. The main result in [17] then concludes that (in a slightly different formulation, where precise degrees rather than expected degrees are specified) the chromatic number 𝜒(𝐺(𝐰))=Θ(𝑑/ln𝑑)=Θ(𝑛^𝛼/ln𝑛) a.a.s. On the other hand, Theorem 1 yields a comparable upper bound warmth(𝐺)≤(1−𝛿)(𝑛^𝛼/ln𝑛)+1.

For dense random graphs, we have the following lower bound.

Theorem 2 (lower bound). For a random graph 𝐺∈𝐺(𝐰), suppose 𝑤²_𝑖/𝑛𝑑=Θ(1) for any 1≤𝑖≤𝑛. Then, for 0<𝛿<1 and 𝑐≥2, one has 𝑤𝑎𝑟𝑚𝑡ℎ(𝐺)≥(1−𝛿)ln_𝑐𝑛𝑎.𝑎.𝑠. (3)

We remark that the above result implies Theorem 3.4 in [5] when 𝑐=2 and 𝑤₁=⋯=𝑤_𝑛=𝑛/2. In view of [2, Theorem 5.1] (as mentioned in Section 1), Theorem 2 provides an alternative approach to obtaining lower bounds for 𝜒(𝐺(𝐰)).

3. Proofs

For a graph 𝐻, let 𝛿(𝐻) denote the minimum degree of 𝐻. For a vertex 𝑣∈𝑉(𝐻), the neighborhood of 𝑣 is denoted by 𝑁(𝑣), and, for a subset 𝐴⊂𝑉(𝐻), the neighborhood of 𝐴 is defined as 𝑁(𝐴)=∪_𝑣∈𝐴𝑁(𝑣). A collection {𝐴_𝑖}^𝑘_𝑖=1 of subsets of 𝐻 is called a 𝑑-stable family if for any 1≤𝑖≤𝑘 there are 𝐴_𝑖1,…,𝐴_𝑖𝑑⊂𝑉(𝐻) such that ∩^𝑑_𝑗=1𝑁(𝐴_𝑖𝑗)=𝐴_𝑖.

We will need the following lemma to prove Theorem 1.

Lemma 3 (see [2]). Given a graph 𝐻 and a natural number 𝑑≥1, 𝐻 is not (𝑑+2)-warm if and only if there is a 𝑑-stable family of subsets of 𝐻.

Proof of Theorem 1. Since 𝜔(𝑛)→∞, for 0<𝛿<1, we have 𝑃(𝛿(𝐺)≥(1−𝛿/2)𝜔(𝑛))→1 as 𝑛→∞ using a concentration inequality [7]. Set 𝑠=(1−𝛿)𝜔(𝑛). We will prove that warmth(𝐺)≤𝑠+1.
Now consider 𝑉(𝐺) consisting of all singleton vertices of 𝐺. A vertex set {𝑣₁,…,𝑣_𝑠} is called an 𝑠-representative [5] of 𝑣∈𝑉(𝐺) if 𝑣₁,…,𝑣_𝑠∈𝑁(𝑣) such that all of them are not in the neighborhood 𝑁(𝑢) for any vertex 𝑢≠𝑣. Therefore, by Lemma 3, it suffices to prove that every vertex 𝑣 has an 𝑠-representative.
Suppose 𝑣₁,…,𝑣_𝑠∈𝑁(𝑣). Let 𝐴(𝑣)(𝑣₁,…,𝑣_𝑠) denote the event that 𝑣₁,…,𝑣_𝑠 are in the neighborhood 𝑁(𝑣) of 𝑣. Hence, 𝑃𝐴𝑣_𝑖𝑣₁,…,𝑣_𝑠forsome𝑣_𝑖(𝐺)≤∈𝑉_{𝑣𝑖(𝐺)∈𝑉}𝑃𝐴𝑣_𝑖𝑣₁,…,𝑣_𝑠=_{𝑣𝑖(𝐺)}^𝑠_{∈𝑉𝑗=1}𝑝_𝑖𝑗=_{𝑣𝑖∈𝑉(𝐺)}𝑤^𝑠_𝑖∏^𝑠_𝑗=1𝑤_𝑗(𝑛𝑑)^𝑠=∑^𝑛_𝑖=1𝑤^𝑠_𝑖∏^𝑠_𝑗=1𝑤_𝑗(𝑛𝑑)^𝑠≤∑^𝑛_𝑖=1𝑤^𝑠_𝑖𝑚^𝑠(𝑛𝑑)^𝑠. (4)
Recall that |𝑁(𝑣)|≥(1−𝛿/2)𝜔(𝑛) as shown in the beginning of the proof. Let 𝛾=⌊(1−𝛿/2)/(1−𝛿)⌋ and 𝑁₁,…,𝑁_𝛾 be some disjoint subsets of the neighbors of 𝑣  with |𝑁_𝑖|=𝑠 for 1≤𝑖≤𝛾. For 𝑀⊂𝑁(𝑣) and |𝑀|=𝑠, denote by 𝐴(𝑀) the event that 𝑀⊂𝑁(𝑢) for some 𝑢≠𝑣. Therefore, the disjointness and inequality (4) imply 𝑃_{|𝑀|(𝑣)=𝑠,𝑀⊂𝑁}𝐴(𝑀)≤𝑃^𝛾_𝑖=1𝐴𝑁_𝑖≤∑^𝑛_𝑖=1𝑤^𝑠_𝑖^𝛾𝑚^𝑠𝛾(𝑛𝑑)^𝑠𝛾. (5) Thus, the probability that some vertex does not have an 𝑠-representative is bounded from above by _𝑣∈𝑉∑^𝑛_𝑖=1𝑤^𝑠_𝑖^𝛾𝑚^𝑠𝛾(𝑛𝑑)^𝑠𝛾=𝑛∑^𝑛_𝑖=1𝑤^𝑠_𝑖^𝛾𝑚^𝑠𝛾(𝑛𝑑)^𝑠𝛾=𝑂𝑚^2𝛾𝑠𝑛^{1+𝛾−𝛾𝑠}, (6) which tends to zero since 𝑚=𝑂(𝑛^𝛼) with 0<𝛼<1/2 and 𝑠→∞. This completes the proof of Theorem 1.

Proof of Theorem 2. Set 𝑠=(1−𝛿)ln_𝑎𝑛−2. By definition, we need to prove that Hom(𝑇^𝑠,𝐺) contains no cold maps.
In what follows, we proceed with the similar lines of reasoning of Section 5 in [5]. We label the vertices of 𝑠-branching rooted tree 𝑇^𝑠 according to Figure 1 with the root labeled 0 and its children 1,2,…, sequently.
Let 𝑇^𝑠_𝑣 denote the truncated version of 𝑠-branching rooted tree 𝑇^𝑠 with 𝑣 vertices, labeled 0 to 𝑣−1. We suppose 𝑣≡1(mod𝑠) and 𝑇^𝑠_𝑣 has 𝑚  vertices up to level 𝑘  and 𝑣−𝑚 vertices from level 𝑘+1 of  𝑇^𝑠, where 𝑚=1+𝑠+⋯+𝑠^𝑘−1=(𝑠^𝑘−1)/(𝑠−1). Hence, we have 𝑚<𝑣≤𝑠𝑚+1 and 𝑣≡𝑚≡1(mod𝑠). The number of leaves of 𝑇^𝑠_𝑣 can be calculated as 𝑙=𝑣−𝑚+𝑠^𝑘−1−𝑣−𝑚𝑠=(𝑠−1)𝑣+1𝑠. (7) The leaves are labeled as (𝑣−1)/𝑠,(𝑣−1)/𝑠+1,…,𝑣−1. Denote by 𝑆=𝑆(𝑇^𝑠_𝑣) the set of roots and leaves; that is, 𝑆={0,(𝑣−1)/𝑠,(𝑣−1)/𝑠+1,…,𝑣−1}.
For a graph 𝐻 and a function 𝑓∶𝑆→𝑉(𝐻), 𝑓 is said to be 𝐻-extendable if there is a homomorphism 𝜙∶𝑇^𝑠_𝑣→𝐻 such that 𝜙|_𝑆=𝑓. Hence, if every function 𝑓∶𝑆→𝑉(𝐺) is 𝐺-extendable, then Hom(𝑇^𝑠,𝐺) contains no cold maps, and thus the proof will be complete.
Now let ()𝜖=1−1−𝛿ln_𝑐𝑛−2ln_𝑐𝑛, (8) which is bounded away from 0 for large enough 𝑛. We claim the following.
Claim 1. For 𝑖∈𝑆(𝑇^𝑠_𝑣) and 𝑎,𝑏_𝑖∈𝐺, the probability that a function 𝑓 with 𝑓(0)=𝑎 and 𝑓(𝑖)=𝑏_𝑖 is not 𝐺-extendable is at most exp(−𝐶𝑛^𝛽) for some constants 𝐶,𝛽>0.
Proof of Claim 1. We use Janson’s inequality [15] to prove Claim 1. Note that by definition (8) the assumptions 𝑤²_𝑖/𝑛𝑑=Θ(1) for any 1≤𝑖≤𝑛 are equivalent to 𝑤²_min𝑛𝑛𝑑=Ω^{−1/ln𝑐𝑛}=Ω𝑛^{−(1−𝜖)/𝑠},𝑤²_max𝑛𝑛𝑑=𝑂^{−1/ln𝑐𝑛}=𝑂𝑛^{−(1−𝜖)/𝑠}. (9)(10)
Let {𝑐_𝑗}=𝑓(𝑆⧵{𝑎,𝑏_𝑖})⊂𝑉(𝐺). Then, from (9) and (10), the probability that there does not exist such a 𝜙 for a particular choice of {𝑎,𝑏_𝑖,𝑐_𝑗} is at most 1−Ω(𝑛^{−(1−𝜖)(𝑣−1)/𝑠}) and at least 1−𝑂(𝑛^{−(1−𝜖)(𝑣−1)/𝑠}).
Let |𝑉(𝐴(𝐴)∩𝒜=𝐴isaninducedsubgraphof𝐺∣)|=𝑣−𝑙−1,𝑉𝑎,𝑏_𝑖=∅. (11) We have |𝒜|=^{𝑛−𝑙−1}_{𝑣−𝑙−1}. For 𝐴_𝑖∈𝒜, denote by {𝑐^𝑖₁,…,𝑐^𝑖_{𝑣−𝑙−1}} the set of vertices of 𝐴_𝑖 ordered according to its labeling. Denote by 𝐵_𝑖 the event that 𝜙 governed by 𝑐_𝑗=𝑐^𝑖_𝑗 is a homomorphism. Hence, the above discussion implies 𝑃𝐵_𝑖=1−Θ𝑛^{−(1−𝜖)(𝑣−1)/𝑠}, (12) where 𝐵_𝑖 denotes the complement of the event 𝐵_𝑖.
Let 𝐴^′ be a subgraph of 𝐺 induced by 𝑉(𝐴)∪{𝑎,𝑏_𝑖}. Let 𝑖∼𝑗 mean that the edges of 𝐴^′_𝑖 and 𝐴^′_𝑗 have a nontrivial intersection. We apply Janson’s inequality [15] and (12) to get 𝑃_𝑖𝐵_𝑖Δ≤exp−𝜇+min_𝑖𝑃𝐵_𝑖Δ≤exp−𝜇+1−Θ𝑛^{−(1−𝜖)(𝑣−1)/𝑠}, (13) where 𝜇=∑_𝑖𝑃(𝐵_𝑖) and Δ=∑_𝑖∼𝑗𝑃(𝐵_𝑖∩𝐵_𝑗).
From (9) and (12) we have 𝜇=Θ𝑛^{𝑣−𝑙−1}𝑛^{(1−𝜖)(𝑣−1)/𝑠}=Θ𝑛^{−1+(𝜖(𝑣−1)/𝑠)}. (14) To estimate Δ, we denote by 𝑔_𝑘=max_𝑖,𝑗{|𝐸(𝐴^′_𝑖∩𝐴^′_𝑗)|∣|𝑉(𝐴_𝑖∩𝐴_𝑗)|=𝑘} the maximum over all pairs 𝑖,𝑗 of the number of edges that 𝐴_𝑖 and 𝐴_𝑗 intersect in 𝑘 vertices. Since the edge sets of 𝐴^′_𝑖 and 𝐴^′_𝑗 have a nontrivial intersection, we get 𝑘≥1. Hence by (10) 𝑃𝐵_𝑖∩𝐵_𝑗≤𝑤²_max𝑛𝑑^{2(𝑣−1)−𝑔𝑘}=𝑂𝑛^{−(1−𝜖)(2(𝑣−1)−𝑠𝑘)/𝑠}, (15) since 𝑔_𝑘≤𝑠𝑘. Noting that there are _𝑘^{𝑛−𝑙−1𝑛−𝑙−1−𝑘}_{𝑣−𝑙−1−𝑘}^{𝑛−𝑙−1−𝑣+𝑙+1}_{𝑣−𝑙−1−𝑘}=𝑂(𝑛^{2𝑣−2𝑙−2−𝑘}) such pairs of 𝐴_𝑖 and 𝐴_𝑗, we obtain Δ=^{𝑣−𝑙−1}_𝑘=1𝑂𝑛^{2𝑣−2𝑙−2−𝑘}𝑛^{−(1−𝜖)(2(𝑣−1)−𝑠𝑘)/𝑠}=𝑂𝑛^{−2+(𝜖/𝑠)(2𝑣−𝑠−2)}. (16)
We choose 𝑣 such that 𝑣≡𝑚≡1(mod𝑠) and 𝑠/𝜖+1<𝑣<𝑠/𝜖+𝑠+1. Hence, estimates (14) and (16) readily yield Δ1−Θ𝑛^{−(1−𝜖)(𝑣−1)/𝑠}=𝑂𝑛^{−2+(𝜖/𝑠)(2𝑣−𝑠−2)}=𝑜𝜇. (17)
Note that 𝛽=liminf_𝑛→∞(−1+𝜖(𝑣−1)/𝑠)>0 by our choice of 𝑣. Thus, inequality (13) in conjunction with estimates (14) and (17) means that 𝑃_𝑖𝐵_𝑖≤𝑒^{−𝐶𝑛𝛽} (18) for some constant 𝐶>0, which then concludes the proof of the claim.
Since there are at most 𝑛^𝑙+1 choices for {𝑎,𝑏_𝑖}, the probability that a homomorphism 𝜙 (with 𝜙|_𝑆=𝑓) does not exist for at least one choice is at most 𝑛^𝑙+1exp(−𝐶𝑛^𝛽), which tends to zero as 𝑛→∞. Consequently, with probability one, every map 𝑓∶𝑆(𝑇^𝑠_𝑣)→𝑉(𝐺) is 𝐺-extendable for some 𝑣. The proof of Theorem 2 is then complete.

4. Conclusion

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 warmth(𝐺)≤𝜒(𝐺). It is worth noting that Fadnavis and Kahle [5] 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 [5] 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.

Acknowledgments

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.