The chromatic discrepancy of graphs

For a proper vertex coloring cc of a graph GG, let φc(G)φc(G) denote the maximum, over all induced subgraphs HH of GG, the difference between the chromatic number χ(H)χ(H) and the number of colors used by cc to color HH. We define the chromatic discrepancy of a graph GG, denoted by φ(G)φ(G), to be the minimum φc(G)φc(G), over all proper colorings cc of GG. If HH is restricted to only connected induced subgraphs, we denote the corresponding parameter by View the MathML sourceφˆ(G). These parameters are aimed at studying graph colorings that use as few colors as possible in a graph and all its induced subgraphs. We study the parameters φ(G)φ(G) and View the MathML sourceφˆ(G) and obtain bounds on them. We obtain general bounds, as well as bounds for certain special classes of graphs including random graphs. We provide structural characterizations of graphs with φ(G)=0φ(G)=0 and graphs with View the MathML sourceφˆ(G)=0. We also show that computing these parameters is NP-hard

Random Tensors and Planted Cliques

The r-parity tensor of a graph is a generalization of the adjacency matrix, where the tensor's entries denote the parity of the number of edges in subgraphs induced by r distinct vertices. For r=2, it is the adjacency matrix with 1's for edges and -1's for nonedges. It is well-known that the 2-norm of the adjacency matrix of a random graph is O(\sqrt{n}). Here we show that the 2-norm of the r-parity tensor is at most f(r)\sqrt{n}\log^{O(r)}n, answering a question of Frieze and Kannan who proved this for r=3. As a consequence, we get a tight connection between the planted clique problem and the problem of finding a vector that approximates the 2-norm of the r-parity tensor of a random graph. Our proof method is based on an inductive application of concentration of measure

The distribution of clusters in random graphs

AbstractGiven a random graph, we investigate the occurrence of subgraphs especially rich in edges. Specifically, given a ϵ [0,1], a set of k points in a graph G is defined to be an a-cluster of cardinality k if the induced subgraph contains at least ak2 edges, so that in the extreme case a = 1, an a-cluster is the same as a clique. We let G = G(n, p) be a random graph on n vertices with edges chosen independently with probability p. Let W denote the number of a-clusters of cardinality k in G, where k and n tend to infinity so that the expected number λ of a-clusters of cardinality k does not grow or decay too rapidly. We prove that W is asymptotically distributed as Zλ, whose distribution is Poisson with mean λ, which is the same result that Bollobás and Erdös have proved for cliques. In contrast to the situation for cliques (a = 1) however, for all a < 1 the second moment of W blows up, i.e., the expected number of neighbors of a given cluster tends to infinity. Nevertheless, the probability that there exists at least one pair of neighboring clusters tends to zero, and a Poisson approximation for W is valid

Spectral pseudorandomness and the road to improved clique number bounds for Paley graphs

We study subgraphs of Paley graphs of prime order $p$ induced on the sets of vertices extending a given independent set of size $a$ to a larger independent set. Using a sufficient condition proved in the author's recent companion work, we show that a family of character sum estimates would imply that, as $p \to \infty$, the empirical spectral distributions of the adjacency matrices of any sequence of such subgraphs have the same weak limit (after rescaling) as those of subgraphs induced on a random set including each vertex independently with probability $2^{-a}$, namely, a Kesten-McKay law with parameter $2^a$. We prove the necessary estimates for $a = 1$, obtaining in the process an alternate proof of a character sum equidistribution result of Xi (2022), and provide numerical evidence for this weak convergence for $a \geq 2$. We also conjecture that the minimum eigenvalue of any such sequence converges (after rescaling) to the left edge of the corresponding Kesten-McKay law, and provide numerical evidence for this convergence. Finally, we show that, once $a \geq 3$, this (conjectural) convergence of the minimum eigenvalue would imply bounds on the clique number of the Paley graph improving on the current state of the art due to Hanson and Petridis (2021), and that this convergence for all $a \geq 1$ would imply that the clique number is $o(\sqrt{p})$.Comment: 43 pages, 1 table, 6 figure

Detecting High Log-Densities -- an O(n^1/4) Approximation for Densest k-Subgraph

In the Densest k-Subgraph problem, given a graph G and a parameter k, one needs to find a subgraph of G induced on k vertices that contains the largest number of edges. There is a significant gap between the best known upper and lower bounds for this problem. It is NP-hard, and does not have a PTAS unless NP has subexponential time algorithms. On the other hand, the current best known algorithm of Feige, Kortsarz and Peleg, gives an approximation ratio of n^(1/3-epsilon) for some specific epsilon > 0 (estimated at around 1/60). We present an algorithm that for every epsilon > 0 approximates the Densest k-Subgraph problem within a ratio of n^(1/4+epsilon) in time n^O(1/epsilon). In particular, our algorithm achieves an approximation ratio of O(n^1/4) in time n^O(log n). Our algorithm is inspired by studying an average-case version of the problem where the goal is to distinguish random graphs from graphs with planted dense subgraphs. The approximation ratio we achieve for the general case matches the distinguishing ratio we obtain for this planted problem. At a high level, our algorithms involve cleverly counting appropriately defined trees of constant size in G, and using these counts to identify the vertices of the dense subgraph. Our algorithm is based on the following principle. We say that a graph G(V,E) has log-density alpha if its average degree is Theta(|V|^alpha). The algorithmic core of our result is a family of algorithms that output k-subgraphs of nontrivial density whenever the log-density of the densest k-subgraph is larger than the log-density of the host graph.Comment: 23 page