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

We consider the random graph model G(w) for a given expected degree sequence w = (w 1 , w 2 , . . . , 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 onwarmth ofG(w). In particular, theminimum expected degree turns out to be an upper bound of warmth when it tends to infinity and the maximum expected degreem = O(nα) with 0 < α < 1/2.


Introduction
Let  = ((), ()) be a graph with vertex set () and edge set ().For graphs  and , a function  : () → () is said to be a graph homomorphism [1] if it induces a map between edges () → ().Denote by Hom(, ) 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 Hom(  , ) is said to be cold if there is a vertex V of  such that for any  no  ∈ Hom(  , ) agrees with  on the vertices at distance  from the root  but has () = V.We say that  is -warm if Hom( −2 , ) does not contain any cold maps.Moreover, the warmth, warmth(), of  is defined to be the largest  for which  is -warm.By definition, for any finite and connected graph , warmth() ≥ 2 and warmth() = 2 if and only if  is bipartite.
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  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 [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 (w).For a given sequence w = ( 1 ,  2 , . . .,   ), (w) is defined as follows.Each potential edge between vertices V  and V  is chosen with probability   and is independent of other edges, where Here, we assume that  2 max = max   2  < ∑  =1   and define  = ∑  =1   .An immediate consequence of (1) is that the expected degree at a vertex V  is exactly   [7].Hence,  is the expected average degree.
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.
For a sparse random graph (w), we may upper bound its warmth using minimum expected degree.
For dense random graphs, we have the following lower bound.

Proofs
For a graph , let () denote the minimum degree of .For a vertex V ∈ (), the neighborhood of V is denoted by (V), and, for a subset  ⊂ (), the neighborhood of  is defined as () = ∪ V∈ (V).A collection {  }  =1 of subsets of  is called a -stable family if for any 1 ≤  ≤  there are   1 , . . .,    ⊂ () such that ∩  =1 (   ) =   .We will need the following lemma to prove Theorem 1.
Lemma 3 (see [2]).Given a graph  and a natural number  ≥ 1,  is not ( + 2)-warm if and only if there is a -stable family of subsets of .

Now consider 𝑉(𝐺) consisting of all singleton vertices of
such that all of them are not in the neighborhood () for any vertex  ̸ = V.Therefore, by Lemma 3, it suffices to prove that every vertex V has an representative. Suppose Recall that |(V)| ≥ (1 − /2)() as shown in the beginning of the proof.Let  = ⌊(1 − /2)/(1 − )⌋ and  1 , . . .,   be some disjoint subsets of the neighbors of V with |  | =  for 1 ≤  ≤ .For  ⊂ (V) and || = , denote by () the event that  ⊂ () for some  ̸ = V.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 0 <  < 1/2 and  → ∞.This completes the proof of Theorem 1.
In what follows, we proceed with the similar lines of reasoning of Section 5 in [5].We label the vertices of branching rooted tree   according to Figure 1 with the root labeled 0 and its children 1, 2, . .., sequently.
For a graph  and a function  :  → (),  is said to be -extendable if there is a homomorphism  :   V →  such that |  = .Hence, if every function  :  → () is -extendable, then Hom(  , ) contains no cold maps, and thus the proof will be complete.Now let which is bounded away from 0 for large enough .We claim the following.
Since there are at most  +1 choices for {,   }, the probability that a homomorphism  (with |  = ) does not exist for at least one choice is at most  +1 exp(−  ), which tends to zero as  → ∞.Consequently, with probability one, every map  : (  V ) → () is -extendable for some V.The proof of Theorem 2 is then complete.

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() ≤ ().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][19][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.