Edge colorings of graphs on surfaces and star edge colorings of sparse graphs

In my dissertation, I present results on two types of edge coloring problems for graphs. For each surface Σ, we define ∆(Σ) = max{∆(G)| G is a class two graph with maximum degree ∆(G) that can be embedded in Σ}. Hence Vizing’s Planar Graph Conjecture can be restated as ∆(Σ) = 5 if Σ is a sphere. For a surface Σ with characteristic χ(Σ) ≤ 0, it is known ∆(Σ) ≥ H(χ(Σ))−1, where H(χ(Σ)) is the Heawood number of the surface, and if the Euler char- acteristic χ(Σ) ∈ {−7, −6, . . . , −1, 0}, ∆(Σ) is already known. I study critical graphs on general surfaces and show that (1) if G is a critical graph embeddable on a surface Σ with Euler character- istic χ(Σ) ∈ {−6, −7}, then ∆(Σ) = 10, and (2) if G is a critical graph embeddable on a surface Σ with Euler characteristic χ(Σ) ≤ −8, then ∆(G) ≤ H(χ(Σ)) (or H(χ(Σ))+1) for some special families of graphs, namely if the minimum degree is at most 11 or if ∆ is very large et al. As applications, we show that ∆(Σ) ≤ H (χ(Σ)) if χ(Σ) ∈ {−22, −21, −20, −18, −17, −15, . . . , −8}and ∆(Σ) ≤ H (χ(Σ)) + 1 if χ(Σ) ∈ {−53, . . . , 23, −19, −16}. Combining this with [19], it follows that if χ(Σ) = −12 and Σ is orientable, then ∆(Σ) = H(χ(Σ)). A star k-edge-coloring is a proper k-edge-coloring such that every connected bicolored sub- graph is a path of length at most 3. The star chromatic index χ′st(G) of a graph G is the smallest integer k such that G has a star k-edge-coloring. The list star chromatic index ch′st(G) is defined analogously. Bezegova et al. and Deng et al. independently proved that χ′ (T) ≤ 3∆ for anyst 2 tree T with maximum degree ∆. Here, we study the list star edge coloring and give tree-like bounds for (list) star chromatic index of sparse graphs. We show that if mad(G) \u3c 2.4, then χ′ (G)≤3∆+2andifmad(G)\u3c15,thench′ (G)≤3∆+1.Wealsoshowthatforeveryε\u3e0st 2 7 st 2 there exists a constant c(ε) such that if mad(G) \u3c 8 − ε, then ch′ (G) ≤ 3∆ + c(ε). We also3 st 2 find guaranteed substructures of graph with mad(G) \u3c 3∆ − ε which may be of interest in other2 problems for sparse graphs

Simulating sparse Hamiltonians with star decompositions

We present an efficient algorithm for simulating the time evolution due to a sparse Hamiltonian. In terms of the maximum degree d and dimension N of the space on which the Hamiltonian H acts for time t, this algorithm uses (d^2(d+log* N)||Ht||)^{1+o(1)} queries. This improves the complexity of the sparse Hamiltonian simulation algorithm of Berry, Ahokas, Cleve, and Sanders, which scales like (d^4(log* N)||Ht||)^{1+o(1)}. To achieve this, we decompose a general sparse Hamiltonian into a small sum of Hamiltonians whose graphs of non-zero entries have the property that every connected component is a star, and efficiently simulate each of these pieces.Comment: 11 pages. v2: minor correction

Distance-two coloring of sparse graphs

Consider a graph $G = (V, E)$ and, for each vertex $v \in V$, a subset $\Sigma(v)$ of neighbors of $v$. A $\Sigma$-coloring is a coloring of the elements of $V$ so that vertices appearing together in some $\Sigma(v)$ receive pairwise distinct colors. An obvious lower bound for the minimum number of colors in such a coloring is the maximum size of a set $\Sigma(v)$, denoted by $\rho(\Sigma)$. In this paper we study graph classes $F$ for which there is a function $f$, such that for any graph $G \in F$ and any $\Sigma$, there is a $\Sigma$-coloring using at most $f(\rho(\Sigma))$ colors. It is proved that if such a function exists for a class $F$, then $f$ can be taken to be a linear function. It is also shown that such classes are precisely the classes having bounded star chromatic number. We also investigate the list version and the clique version of this problem, and relate the existence of functions bounding those parameters to the recently introduced concepts of classes of bounded expansion and nowhere-dense classes.Comment: 13 pages - revised versio

Induced Ramsey-type theorems

We present a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earlier results of Rodl, Erdos-Hajnal, Promel-Rodl, Nikiforov, Chung-Graham, and Luczak-Rodl. The proofs are based on a simple lemma (generalizing one by Graham, Rodl, and Rucinski) that can be used as a replacement for Szemeredi's regularity lemma, thereby giving much better bounds. The same approach can be also used to show that pseudo-random graphs have strong induced Ramsey properties. This leads to explicit constructions for upper bounds on various induced Ramsey numbers.Comment: 30 page

A sharp threshold for random graphs with a monochromatic triangle in every edge coloring

Let $\R$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the edges of $G$ by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property. Let $G(n,p)$ be the random graph on $n$ vertices with edge probability $p$. We prove that there exists a function $\hat c=\hat c(n)$ with $0 0$, as $n$ tends to infinity Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is of independent interest is a generalization of Szemer\'edi's Regularity Lemma to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.

Color-blind index in graphs of very low degree

Let $c:E(G)\to [k]$ be an edge-coloring of a graph $G$, not necessarily proper. For each vertex $v$, let $\bar{c}(v)=(a_1,\ldots,a_k)$, where $a_i$ is the number of edges incident to $v$ with color $i$. Reorder $\bar{c}(v)$ for every $v$ in $G$ in nonincreasing order to obtain $c^*(v)$, the color-blind partition of $v$. When $c^*$ induces a proper vertex coloring, that is, $c^*(u)\neq c^*(v)$ for every edge $uv$ in $G$, we say that $c$ is color-blind distinguishing. The minimum $k$ for which there exists a color-blind distinguishing edge coloring $c:E(G)\to [k]$ is the color-blind index of $G$, denoted $\operatorname{dal}(G)$. We demonstrate that determining the color-blind index is more subtle than previously thought. In particular, determining if $\operatorname{dal}(G) \leq 2$ is NP-complete. We also connect the color-blind index of a regular bipartite graph to 2-colorable regular hypergraphs and characterize when $\operatorname{dal}(G)$ is finite for a class of 3-regular graphs.Comment: 10 pages, 3 figures, and a 4 page appendi

Near-colorings: non-colorable graphs and NP-completeness

A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitioned into subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices of V_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focus on complexity aspects of such colorings when l=2,3. More precisely, we prove that, for any fixed integers k,j,g with (k,j) distinct form (0,0) and g >= 3, either every planar graph with girth at least g is (k,j)-colorable or it is NP-complete to determine whether a planar graph with girth at least g is (k,j)-colorable. Also, for any fixed integer k, it is NP-complete to determine whether a planar graph that is either (0,0,0)-colorable or non-(k,k,1)-colorable is (0,0,0)-colorable. Additionally, we exhibit non-(3,1)-colorable planar graphs with girth 5 and non-(2,0)-colorable planar graphs with girth 7