Induced Turán Numbers

Let $\Delta(G)$ be the maximum degree of a graph $G$. Brooks' theorem states that the only connected graphs with chromatic number $\chi(G)=\Delta(G)+1$ are complete graphs and odd cycles. We prove a fractional analogue of Brooks' theorem in this paper. Namely, we classify all connected graphs $G$ such that the fractional chromatic number $\chi_f(G)$ is at least $\Delta(G)$. These graphs are complete graphs, odd cycles, $C^2_8$, $C_5\boxtimes K_2$, and graphs whose clique number $\omega(G)$ equals the maximum degree $\Delta(G)$. Among the two sporadic graphs, the graph $C^2_8$ is the square graph of cycle $C_8$ while the other graph $C_5\boxtimes K_2$ is the strong product of $C_5$ and $K_2$. In fact, we prove a stronger result; if a connected graph $G$ with $\Delta(G)\geq 4$ is not one of the graphs listed above, then we have $\chi_f(G)\leq \Delta(G)- 2/67$.

Group-testing refers to the problem of identifying (with high probability) a (small) subset of $D$ defectives from a (large) set of $N$ items via a “small” number of “pooled” tests (i.e., tests that have a positive outcome if at least one of the items being tested in the pool is defective, else have a negative outcome). For ease of presentation in this paper, we focus on the regime when $D = \mathcal {O}(N^{1- {\delta }})$ for some $ {\delta }> 0$ . The tests may be noiseless or noisy , and the testing procedure may be adaptive (the pool defining a test may depend on the outcome of a previous test), or non-adaptive (each test is performed independent of the outcome of other tests). A rich body of the literature demonstrates that $\Theta (D\log (N))$ tests are information-theoretically necessary and sufficient for the group-testing problem, and provides algorithms that achieve this performance. However, it is only recently that reconstruction algorithms with computational complexities that are sub-linear in $N$ have started being investigated. In the scenario with adaptive tests with noisy outcomes, we present the first scheme that is simultaneously order-optimal (up to small constant factors) in both the number of tests and the decoding complexity ( $ \mathcal {O}\left ({D\log (N)}\right )$ in both the performance metrics). The total number of stages of our adaptive algorithm is “small” ( $ \mathcal {O}\left ({\log (D)}\right )$ ). Similarly, in the scenario with non-adaptive tests with noisy outcomes, we present the first scheme that is simultaneously near-optimal in both the number of tests and the decoding complexity (via an algorithm that requires $ \mathcal {O}\left ({D\log (D)\log (N)}\right )$ tests and has a decoding complexity of ${\mathcal{ O}}(D(\log N+\log ^{2}D))$ . Finally, we present an adaptive algorithm that only requires two stages, and for which both the number of tests and the decoding complexity scale as ${\mathcal{ O}}(D(\log N+\log ^{2}D))$ . For all three settings, the probability of error of our algorithms scales as $ \mathcal {O}\left ({1/(poly(D)}\right )$ . For each of the statements mentioned earlier about the order of the number of measurements, decoding complexity, and probability of error, we provide explicitly computed “small” universal factors in our theorem statements.

A graph is ‐colorable if its vertex set can be partitioned into sets , such that for each , the subgraph of induced by has maximum degree at most . The Four Color Theorem states that every planar graph is ‐colorable, and a classical result of Cowen, Cowen, and Woodall shows that every planar graph is ‐colorable. In this paper, we extend both of these results to graphs on surfaces. Namely, we show that every graph embeddable on a surface of Euler genus is ‐colorable and ‐colorable. Moreover, these graphs are also ‐colorable and ‐colorable. We also prove that every triangle‐free graph that is embeddable on a surface of Euler genus is ‐colorable. This is an extension of Grötzsch's Theorem, which states that triangle‐free planar graphs are ‐colorable. Finally, we prove that every graph of girth at least 7 that is embeddable on a surface of Euler genus is ‐colorable. All these results are best possible in several ways as the girth condition is sharp, the constant maximum degrees cannot be improved, and the bounds on the maximum degrees depending on are tight up to a constant multiplicative factor.

The gamma graph of a graph

In Hell et al. (2004), we have previously observed that, in a chordal graph G, the maximum number of independent r -cliques (i.e., of vertex disjoint subgraphs of G, each isomorphic to Kr , with no edges joining any two of the subgraphs) equals the minimum number of cliques of G that meet all the r -cliques of G. When r = 1, this says that chordal graphs have independence number equal to the clique covering number. When r = 2, this is equivalent to a result of Cameron (1989), and it implies a well known forbidden subgraph characterization of split graphs. In this paper we consider a weighted version of the above min-max optimization problem. Given a chordal graph G, with a nonnegative weight for each r -clique in G ,w eseek a set of independent r -cliques with maximum total weight. We present two algorithms to solve this problem, based on the principle of complementary slackness. The first one operates on a graph derived from G, and is an adaptation of an algorithm of Farber (1982). The second one improves the performance by reducing the number of constraints of the linear programs. Both results produce a min-max relation. When the algorithms are specialized to the situation in which all the r -cliques have the same weight, we simplify the algorithms reported in Hell et al. (2004), although these simpler algorithms are not as efficient. As a byproduct, we also obtain new elementary proofs of the above min-max result.