Coloring Graphs with Forbidden Minors

Hadwiger's conjecture from 1943 states that for every integer $t\ge1$, every graph either can be $t$-colored or has a subgraph that can be contracted to the complete graph on $t+1$ vertices. As pointed out by Paul Seymour in his recent survey on Hadwiger's conjecture, proving that graphs with no $K_7$ minor are $6$-colorable is the first case of Hadwiger's conjecture that is still open. It is not known yet whether graphs with no $K_7$ minor are $7$-colorable. Using a Kempe-chain argument along with the fact that an induced path on three vertices is dominating in a graph with independence number two, we first give a very short and computer-free proof of a recent result of Albar and Gon\c{c}alves and generalize it to the next step by showing that every graph with no $K_t$ minor is $(2t-6)$-colorable, where $t\in\{7,8,9\}$. We then prove that graphs with no $K_8^-$ minor are $9$-colorable and graphs with no $K_8^=$ minor are $8$-colorable. Finally we prove that if Mader's bound for the extremal function for $K_p$ minors is true, then every graph with no $K_p$ minor is $(2t-6)$-colorable for all $p\ge5$. This implies our first result. We believe that the Kempe-chain method we have developed in this paper is of independent interest

Coloring Graphs with Forbidden Minors

A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. My research is motivated by the famous Hadwiger\u27s Conjecture from 1943 which states that every graph with no Kt-minor is (t − 1)-colorable. This conjecture has been proved true for t ≤ 6, but remains open for all t ≥ 7. For t = 7, it is not even yet known if a graph with no K7-minor is 7-colorable. We begin by showing that every graph with no Kt-minor is (2t − 6)- colorable for t = 7, 8, 9, in the process giving a shorter and computer-free proof of the known results for t = 7, 8. We also show that this result extends to larger values of t if Mader\u27s bound for the extremal function for Kt-minors is true. Additionally, we show that any graph with no K−8 - minor is 9-colorable, and any graph with no K=8-minor is 8-colorable. The Kempe-chain method developed for our proofs of the above results may be of independent interest. We also use Mader\u27s H-Wege theorem to establish some sufficient conditions for a graph to contain a K8-minor. Another motivation for my research is a well-known conjecture of Erdos and Lovasz from 1968, the Double-Critical Graph Conjecture. A connected graph G is double-critical if for all edges xy ∈ E(G), χ(G−x−y) = χ(G)−2. Erdos and Lovasz conjectured that the only double-critical t-chromatic graph is the complete graph Kt. This conjecture has been show to be true for t ≤ 5 and remains open for t ≥ 6. It has further been shown that any non-complete, double-critical, t-chromatic graph contains Kt as a minor for t ≤ 8. We give a shorter proof of this result for t = 7, a computer-free proof for t = 8, and extend the result to show that G contains a K9-minor for all t ≥ 9. Finally, we show that the Double-Critical Graph Conjecture is true for double-critical graphs with chromatic number t ≤ 8 if such graphs are claw-free

Bipartite Minors

We introduce a notion of bipartite minors and prove a bipartite analog of Wagner's theorem: a bipartite graph is planar if and only if it does not contain $K_{3,3}$ as a bipartite minor. Similarly, we provide a forbidden minor characterization for outerplanar graphs and forests. We then establish a recursive characterization of bipartite $(2,2)$-Laman graphs --- a certain family of graphs that contains all maximal bipartite planar graphs.Comment: 9 page

Finding Cycles and Trees in Sublinear Time

We present sublinear-time (randomized) algorithms for finding simple cycles of length at least $k\geq 3$ and tree-minors in bounded-degree graphs. The complexity of these algorithms is related to the distance of the graph from being $C_k$-minor-free (resp., free from having the corresponding tree-minor). In particular, if the graph is far (i.e., $\Omega(1)$-far) {from} being cycle-free, i.e. if one has to delete a constant fraction of edges to make it cycle-free, then the algorithm finds a cycle of polylogarithmic length in time \tildeO(\sqrt{N}), where $N$ denotes the number of vertices. This time complexity is optimal up to polylogarithmic factors. The foregoing results are the outcome of our study of the complexity of {\em one-sided error} property testing algorithms in the bounded-degree graphs model. For example, we show that cycle-freeness of $N$-vertex graphs can be tested with one-sided error within time complexity \tildeO(\poly(1/\e)\cdot\sqrt{N}). This matches the known $\Omega(\sqrt{N})$ query lower bound, and contrasts with the fact that any minor-free property admits a {\em two-sided error} tester of query complexity that only depends on the proximity parameter \e. For any constant $k\geq3$, we extend this result to testing whether the input graph has a simple cycle of length at least $k$. On the other hand, for any fixed tree $T$, we show that $T$-minor-freeness has a one-sided error tester of query complexity that only depends on the proximity parameter \e. Our algorithm for finding cycles in bounded-degree graphs extends to general graphs, where distances are measured with respect to the actual number of edges. Such an extension is not possible with respect to finding tree-minors in $o(\sqrt{N})$ complexity.Comment: Keywords: Sublinear-Time Algorithms, Property Testing, Bounded-Degree Graphs, One-Sided vs Two-Sided Error Probability Updated versio

On First-Order Definable Colorings

We address the problem of characterizing $H$-coloring problems that are first-order definable on a fixed class of relational structures. In this context, we give several characterizations of a homomorphism dualities arising in a class of structure

Span programs and quantum algorithms for st-connectivity and claw detection

We introduce a span program that decides st-connectivity, and generalize the span program to develop quantum algorithms for several graph problems. First, we give an algorithm for st-connectivity that uses O(n d^{1/2}) quantum queries to the n x n adjacency matrix to decide if vertices s and t are connected, under the promise that they either are connected by a path of length at most d, or are disconnected. We also show that if T is a path, a star with two subdivided legs, or a subdivision of a claw, its presence as a subgraph in the input graph G can be detected with O(n) quantum queries to the adjacency matrix. Under the promise that G either contains T as a subgraph or does not contain T as a minor, we give O(n)-query quantum algorithms for detecting T either a triangle or a subdivision of a star. All these algorithms can be implemented time efficiently and, except for the triangle-detection algorithm, in logarithmic space. One of the main techniques is to modify the st-connectivity span program to drop along the way "breadcrumbs," which must be retrieved before the path from s is allowed to enter t.Comment: 18 pages, 4 figure