There are several graphs H for which the precise structure of graphs that do not contain a minor isomorphic to H is known. In particular, such structure theorems are known for K5 [13], V8 [8] and [7], the cube [3], the octahedron [4], and several others. Such characterizations can often be very useful, e.g., Hadwiger’s conjecture for k = 4 is verified by using the structure for K5-free graphs, ...

The planar Ramsey number PR (H1, H2) is the smallest integer n such that any planar graph on n vertices contains a copy of H1 or its complement contains a copy of H2. It is known that the Ramsey number R(K4 − e, K6) = 21, and the planar Ramsey numbers PR(K4 − e, Kl) for l ≤ 5 are known. In this paper, we give the lower bounds on PR (K4 − e, Kl) and determine the exact value of PR (K4 − e, K6).

Klotho (KL) encodes a single-pass transmembrane protein and is predominantly expressed in the kidney, parathyroid glands, and choroid plexus. Genetic studies on the KL gene have revealed that DNA hypermethylation is one of the major risk factors for aging, diseases, and cancer. Besides, KL exerts anti-inflammatory and anti-tumor effects by regulating signaling pathways and the expression of tar...

In this paper, we consider the circular chromatic number c (G) of series-parallel graphs G. It is well known that series-parallel graphs have chromatic number at most 3. Hence their circular chromatic number is also at most 3. If a series-parallel graph G contains a triangle , then both the chromatic number and the circular chromatic number of G are indeed equal to 3. We shall show that if a se...

This work provides a Linear Programming-based Polynomial Time Approximation Scheme (PTAS) for two classical NP-hard problems on graphs when the input graph is guaranteed to be planar, or more generally Minor Free. The algorithm applies a sufficiently large number (some function of 1/ when 1+ approximation is required) of rounds of the so-called Sherali-Adams Lift-and-Project system. needed to o...

‎Let $R$ be a commutative ring with non-zero identity. ‎We describe all $C_3$‎- ‎and $C_4$-free intersection graph of non-trivial ideals of $R$ as well as $C_n$-free intersection graph when $R$ is a reduced ring. ‎Also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎Finally, ‎we shall prove that almost all Artin rings $R$ have Hamiltonian intersection graphs. ...

A graph H is an induced minor of a G if can be obtained from by vertex deletions and edge contractions. We show that there function f(k,d)=O(k10+2d5) so has treewidth at least f(k,d) maximum degree most d, then it contains k×k-grid as minor. This proves the conjecture Aboulker, Adler, Kim, Sintiari, Trotignon (2021) [1] any with large bounded wall or line subgraph. It also implies for fixed pla...

We prove a number of results around kernelization of problems parameterized by the size of a given vertex cover of the input graph. We provide three sets of simple general conditions characterizing problems admitting kernels of polynomial size. Our characterizations not only give generic explanations for the existence of many known polynomial kernels for problems like q-Coloring, Odd Cycle Tran...

Motivated by a problem in large scale climate data analysis, we consider the problem of maximum a posteriori (MAP) inference in graphical models with millions of nodes. While progress has been made in recent years, existing MAP inference algorithms are inherently sequential and hence do not scale well. In this paper, we present a parallel MAP inference algorithm called KL-ADM based on two ideas...