Given a group $\Gamma$ acting on a set $X$, a $k$-coloring $\phi:X\to\{1,\dots,k\}$ of $X$ is distinguishing with respect to $\Gamma$ if the only $\gamma\in \Gamma$ that fixes $\phi$ is the identity action. The distinguishing number of the action $\Gamma$, denoted $D_{\Gamma}(X)$, is then the smallest positive integer $k$ such that there is a distinguishing $k$-coloring of $X$ with respect to $\Gamma$. This notion has been studied in a number of settings, but by far the largest body of work has been concerned with finding the distinguishing number of the action of the automorphism group of a graph $G$ upon its vertex set, which is referred to as the distinguishing number of $G$. The distinguishing number of a group action is a measure of how difficult it is to break all of the permutations arising from that action. In this paper, we aim to further differentiate the resilience of group actions with the same distinguishing number. In particular, we introduce a precoloring extension framework to address this issue. A set $S \subseteq X$ is a fixing set for $\Gamma$ if for every non-identity element $\gamma \in \Gamma$ there is an element $s \in S$ such that $\gamma(s) \neq s$. The distinguishing extension number $\operatorname{ext}_D(X,\Gamma;k)$ is the minimum number $m$ such that for all fixing sets $W \subseteq X$ with $|W| \geq m$, every $k$-coloring $c : X \setminus W \to [k]$ can be extended to a $k$-coloring that distinguishes $X$. In this paper, we prove that $\operatorname{ext}_D(\mathbb{R},\operatorname{Aut}(\mathbb{R}),2) =4$, where $\operatorname{Aut}(\mathbb{R})$ is comprised of compositions of translations and reflections. We also consider the distinguishing extension number of the circle and (finite) cycles, obtaining several exact results and bounds.

The strong chromatic index of a graph $G$, denoted $\chi'_s(G)$, is the least number of colors needed to edge-color $G$ so that edges at distance at most two receive distinct colors. The strong list chromatic index, denoted $\chi'_{s,\ell}(G)$, is the least integer $k$ such that if arbitrary lists of size $k$ are assigned to each edge then $G$ can be edge-colored from those lists where edges at distance at most two receive distinct colors.We use the discharging method, the Combinatorial Nullstellensatz, and computation to show that if $G$ is a subcubic planar graph with ${\rm girth}(G) \geq 41$ then $\chi'_{s,\ell}(G) \leq 5$, answering a question of Borodin and Ivanova [Precise upper bound for the strong edge chromatic number of sparse planar graphs, Discuss. Math. Graph Theory, 33(4), (2014) 759--770]. We further show that if $G$ is a subcubic planar graph and ${\rm girth}(G) \geq 30$, then $\chi_s'(G) \leq 5$, improving a bound from the same paper.Finally, if $G$ is a planar graph with maximum degree at most four and ${\rm girth}(G) \geq 28$, then $\chi'_s(G)N \leq 7$, improving a more general bound of Wang and Zhao from [Odd graphs and its applications to the strong edge coloring, Applied Mathematics and Computation, 325 (2018), 246-251] in this case.

A star k-coloring is a proper k-coloring where the union of two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an I,F-partition. We use a combination of potential functions and discharging to prove that every graph with maximum average degree less than 5 2 has an I,F-partition, which is sharp and answers a question of Cranston and West [A guide to the discharging method, arXiv:1306.4434]. This result implies that planar graphs of girth at least 10 are star 4-colorable, improving upon previous results of Bu, Cranston, Montassier, Raspaud, and Wang [Star coloring of sparse graphs, J. Graph Theory 62 (2009), 201-219].

We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Turán-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.

List coloring generalizes graph coloring by requiring the color of a vertex to be selected from a list of colors specific to that vertex. One refinement of list coloring, called choosability with separation , requires that the intersection of adjacent lists is sufficiently small. We introduce a new refinement, called choosability with union separation , where we require that the union of adjacent lists is sufficiently large. For t ≥ k , a ( k , t ) -list assignment is a list assignment L where | L ( v ) | ≥ k for all vertices v and | L ( u ) ∪ L ( v ) | ≥ t for all edges u v . A graph is ( k , t ) - choosable if there is a proper coloring for every ( k , t ) -list assignment. We explore this concept through examples of graphs that are not ( k , t ) -choosable, demonstrating sparsity conditions that imply a graph is ( k , t ) -choosable, and proving that all planar graphs are ( 3 , 11 ) -choosable and ( 4 , 9 ) -choosable.

Let c:E(G)→[k] be an edge-coloring of a graph G, not necessarily proper. For each vertex v, let c̄(v)=(a1,…,ak), where ai is the number of edges incident to v with color i. Reorder c̄(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)≠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)→[k] is the color-blind index of G, denoted dal(G). We demonstrate that determining the color-blind index is more subtle than previously thought. In particular, determining if dal(G)≤2 is NP-complete. We also connect the color-blind index of a regular bipartite graph to 2-colorable regular hypergraphs and characterize when dal(G) is finite for a class of 3-regular graphs.

All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any \(\ell \in \{3,4,5,6,7\}\), a planar graph is 4-choosable if it is \(\ell \)-cycle-free. In terms of constraining the list assignment, one refinement of k-choosability is choosability with separation. A graph is (k, s)-choosable if the graph is colorable from lists of size k where adjacent vertices have at most s common colors in their lists. Every planar graph is (4, 1)-choosable, but there exist planar graphs that are not (4, 3)-choosable. It is an open question whether planar graphs are always (4, 2)-choosable. A chorded \(\ell \)-cycle is an \(\ell \)-cycle with one additional edge. We demonstrate for each \(\ell \in \{5,6,7\}\) that a planar graph is (4, 2)-choosable if it does not contain chorded \(\ell \)-cycles.

A star $k$-coloring is a proper $k$-coloring where the union of two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an I,F-partition. We use a combination of potential functions and discharging to prove that every graph with maximum average degree less than $\frac{5}{2}$ has an I,F-partition, which is sharp and answers a question of Cranston and West [A guide to the discharging method, arXiv:1306.4434]. This result implies that planar graphs of girth at least 10 are star 4-colorable, improving upon previous results of Bu, Cranston, Montassier, Raspaud, and Wang [Star coloring of sparse graphs, J. Graph Theory 62 (2009), 201-219].

In this paper, we investigate the anti-Ramsey (more precisely, anti-van der Waerden) properties of arithmetic progressions. For positive integers $n$ and $k$, the expression $aw([n],k)$ denotes the smallest number of colors with which the integers $\{1,\ldots,n\}$ can be colored and still guarantee there is a rainbow arithmetic progression of length $k$. We establish that $aw([n],3)=\Theta(\log n)$ and $aw([n],k)=n^{1-o(1)}$ for $k\geq 4$. For positive integers $n$ and $k$, the expression $aw(Z_n,k)$ denotes the smallest number of colors with which elements of the cyclic group of order $n$ can be colored and still guarantee there is a rainbow arithmetic progression of length $k$. In this setting, arithmetic progressions can wrap around, and $aw(Z_n,3)$ behaves quite differently from $aw([n],3)$, depending on the divisibility of $n$. As shown in [Jungi\'c et al., \textit{Combin. Probab. Comput.}, 2003], $aw(Z_{2^m},3) = 3$ for any positive integer $m$. We establish that $aw(Z_n,3)$ can be computed from knowledge of $aw(Z_p,3)$ for all of the prime factors $p$ of $n$. However, for $k\geq 4$, the behavior is similar to the previous case, that is, $aw(Z_n,k)=n^{1-o(1)}$.

Given $k\ge 1$, a $k$-proper partition of a graph $G$ is a partition ${\mathcal P}$ of $V(G)$ such that each part $P$ of ${\mathcal P}$ induces a $k$-connected subgraph of $G$. We prove that if $G$ is a graph of order $n$ such that $\delta(G)\ge \sqrt{n}$, then $G$ has a $2$-proper partition with at most $n/\delta(G)$ parts. The bounds on the number of parts and the minimum degree are both best possible. We then prove that If $G$ is a graph of order $n$ with minimum degree $\delta(G)\ge\sqrt{c(k-1)n}$, where $c=\frac{2123}{180}$, then $G$ has a $k$-proper partition into at most $\frac{cn}{\delta(G)}$ parts. This improves a result of Ferrara, Magnant and Wenger [Conditions for Families of Disjoint $k$-connected Subgraphs in a Graph, Discrete Math. 313 (2013), 760--764] and both the degree condition and the number of parts are best possible up to the constant $c$.

A distance graph is an undirected graph on the integers where two integers are adjacent if their difference is in a prescribed distance set. The independence ratio of a distance graph $G$ is the maximum density of an independent set in $G$. Lih, Liu, and Zhu [Star extremal circulant graphs, SIAM J. Discrete Math. 12 (1999) 491--499] showed that the independence ratio is equal to the inverse of the fractional chromatic number, thus relating the concept to the well studied question of finding the chromatic number of distance graphs. We prove that the independence ratio of a distance graph is achieved by a periodic set, and we present a framework for discharging arguments to demonstrate upper bounds on the independence ratio. With these tools, we determine the exact independence ratio for several infinite families of distance sets of size three, determine asymptotic values for others, and present several conjectures.

We study choosability with separation which is a constrained version of list coloring of graphs. A (k,d) ‐list assignment L of a graph G is a function that assigns to each vertex v a list L(v) of at least k colors and for any adjacent pair xy , the lists L(x) and L(y) share at most d colors. A graph G is (k,d) ‐choosable if there exists an L‐coloring of G for every (k,d) ‐list assignment L. This concept is also known as choosability with separation. We prove that planar graphs without 4‐cycles are (3, 1)‐choosable and that planar graphs without 5‐ and 6‐cycles are (3, 1)‐choosable. In addition, we give an alternative and slightly stronger proof that triangle‐free planar graphs are (3, 1)‐choosable.

For a $k$-uniform hypergraph $G$ with vertex set $\{1,\ldots,n\}$, the ordered Ramsey number $\operatorname{OR}_t(G)$ is the least integer $N$ such that every $t$-coloring of the edges of the complete $k$-uniform graph on vertex set $\{1,\ldots,N\}$ contains a monochromatic copy of $G$ whose vertices follow the prescribed order. Due to this added order restriction, the ordered Ramsey numbers can be much larger than the usual graph Ramsey numbers. We determine that the ordered Ramsey numbers of loose paths under a monotone order grows as a tower of height one less than the maximum degree. We also extend theorems of Conlon, Fox, Lee, and Sudakov [Ordered Ramsey numbers, arXiv:1410.5292] on the ordered Ramsey numbers of 2-uniform matchings to provide upper bounds on the ordered Ramsey number of $k$-uniform matchings under certain orderings.

We introduce an ordered version of Ramsey numbers for hypergraphs using linearly ordered vertex sets. In this model, we obtain bounds on the ordered Ramsey numbers of the k-uniform hypergraph whose edge set consists of the sets of k consecutive vertices in the linear order on the vertex set. The upper and lower bounds are towers of the same height. We apply these bounds to study the minimum number of interval graphs whose union is the line graph of the n-vertex complete graph, proving the conjecture of Heldt, Knauer, and Ueckerdt that this number grows with n. In fact, the growth rate is between Ω( log logn log log logn) and O(log log n).

Any finite group can be encoded as the automorphism group of an unlabeled simple graph. Recently Hartke, Kolb, Nishikawa, and Stolee (2010) demonstrated a construction that allows any ordered pair of finite groups to be represented as the automorphism group of a graph and a vertex deleted subgraph. In this note, we describe a generalized scenario as a game between a player and an adversary: an adversary provides a list of finite groups and a number of rounds. The player constructs a graph with automorphism group isomorphic to the first group. In the following rounds, the adversary selects a group and the player deletes a vertex such that the automorphism group of the corresponding vertex-deleted subgraph is isomorphic to the selected group. We provide a construction that allows the player to appropriately respond to any sequence of challenges from the adversary.

The Manickam-Miklos-Singhi conjecture states that when n=4k, every multiset of n real numbers with nonnegative total sum has at least (n-1k-1)k-subsets with nonnegative sum. We develop a branching strategy using a linear programming formulation to show that verifying the conjecture for fixed values of k is a finite problem. To improve our search, we develop a zero-error randomized propagation algorithm. Using implementations of these algorithms, we verify a stronger form of the conjecture for all k@?7.

Discharging arguments demonstrate a connection between local structure and global averages. This makes it an effective tool for proving lower bounds on the density of special sets in infinite grids. However, the minimum density of an identifying code in the hexagonal grid remains open, with an upper bound of $\frac{3}{7} \approx 0.428571$ and a lower bound of $\frac{5}{12}\approx 0.416666$. We present a new, experimental framework for producing discharging arguments using an algorithm. This algorithm replaces the lengthy case analysis of human-written discharging arguments with a linear program that produces the best possible lower bound using the specified set of discharging rules. We use this framework to present a lower bound of $\frac{23}{55} \approx 0.418181$ on the density of an identifying code in the hexagonal grid, and also find several sharp lower bounds for variations on identifying codes in the hexagonal, square, and triangular grids.

A coloring of the vertices of a graph G is said to be distinguishing} provided no nontrivial automorphism of G preserves all of the vertex colors. The distinguishing number of G, D(G), is the minimum number of colors in a distinguishing coloring of G. The distinguishing chromatic number of G, chi_D(G), is the minimum number of colors in a distinguishing coloring of G that is also a proper coloring. Recently the notion of a distinguishing coloring was extended to that of a list distinguishing coloring. Given an assignment L= {L(v) : v in V(G)} of lists of available colors to the vertices of G, we say that G is (properly) L-distinguishable if there is a (proper) distinguishing coloring f of G such that f(v) is in L(v) for all v. The list distinguishing number of G, D_l(G), is the minimum integer k such that G is L-distinguishable for any list assignment L with |L(v)| = k for all v. Similarly, the list distinguishing chromatic number of G, denoted chi_{D_l}(G) is the minimum integer k such that G is properly L-distinguishable for any list assignment L with |L(v)| = k for all v. In this paper, we study these distinguishing parameters for trees, and in particular extend an enumerative technique of Cheng to show that for any tree T, D_l(T) = D(T), chi_D(T)=chi_{D_l}(T), and chi_D(T) <= D(T) + 1.

Let f(n,p) denote the maximum number of edges in a graph having n vertices and exactly p perfect matchings. For fixed p, Dudek and Schmitt showed that f(n,p)=n2/4+cp for some constant cp when n is at least some constant np. For p6, they also determined cp and np. For fixed p, we show that the extremal graphs for all n are determined by those with O(p) vertices. As a corollary, a computer search determines cp and np for p10. We also present lower bounds on f(n,p) proving that cp>0 for p2 (as conjectured by Dudek and Schmitt), and we conjecture an upper bound on f(n,p). Our structural results are based on Lovasz's Cathedral Theorem. (C) 2012 Wiley Periodicals, Inc.