Coloring a graph optimally with two colors

AbstractLet G be a graph with point set V. A (2-)coloring of G is a map of V to {red, white}. An error occurs whenever the two endpoints of a line have the same color. An optimal coloring of G is a coloring of G for which the number of errors is minimum. The minimum number of errors is denoted by γ(G), we derive upper and lower bounds for γ(G) and prove that if G is a graph with n points and m lines, then max{0, m−⌊14n2⌋}⩽γ(G)⩽⌊12m−14(h(m)−1)⌋, where h(m)=min{n¦m⩽(n2)}. The lower bound is sharp, and for infinitely many values of m the upper bound is attained for all sufficiently large n

Online Coloring of Bipartite Graphs with and without Advice

In the online version of the well-known graph coloring problem, the vertices appear one after the other together with the edges to the already known vertices and have to be irrevocably colored immediately after their appearance. We consider this problem on bipartite, i.e., two-colorable graphs. We prove that at least ⌊1.13746⋅log2(n)−0.49887⌋ colors are necessary for any deterministic online algorithm to be able to color any given bipartite graph on n vertices, thus improving on the previously known lower bound of ⌊log2 n⌋+1 for sufficiently large n. Recently, the advice complexity was introduced as a method for a fine-grained analysis of the hardness of online problems. We apply this method to the online coloring problem and prove (almost) tight linear upper and lower bounds on the advice complexity of coloring a bipartite graph online optimally or using 3 colors. Moreover, we prove that $O(\sqrt{n})$ advice bits are sufficient for coloring any bipartite graph on n vertices with at most ⌈log2 n⌉ colors

Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs

A path in an edge-colored graph $G$ is rainbow if no two edges of it are colored the same. The graph $G$ is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph $G$ is strongly rainbow-connected. The minimum number of colors needed to make $G$ rainbow-connected is known as the rainbow connection number of $G$, and is denoted by $\text{rc}(G)$. Similarly, the minimum number of colors needed to make $G$ strongly rainbow-connected is known as the strong rainbow connection number of $G$, and is denoted by $\text{src}(G)$. We prove that for every $k \geq 3$, deciding whether $\text{src}(G) \leq k$ is NP-complete for split graphs, which form a subclass of chordal graphs. Furthermore, there exists no polynomial-time algorithm for approximating the strong rainbow connection number of an $n$-vertex split graph with a factor of $n^{1/2-\epsilon}$ for any $\epsilon > 0$ unless P = NP. We then turn our attention to block graphs, which also form a subclass of chordal graphs. We determine the strong rainbow connection number of block graphs, and show it can be computed in linear time. Finally, we provide a polynomial-time characterization of bridgeless block graphs with rainbow connection number at most 4.Comment: 13 pages, 3 figure

On the phase transitions of graph coloring and independent sets

We study combinatorial indicators related to the characteristic phase transitions associated with coloring a graph optimally and finding a maximum independent set. In particular, we investigate the role of the acyclic orientations of the graph in the hardness of finding the graph's chromatic number and independence number. We provide empirical evidence that, along a sequence of increasingly denser random graphs, the fraction of acyclic orientations that are `shortest' peaks when the chromatic number increases, and that such maxima tend to coincide with locally easiest instances of the problem. Similar evidence is provided concerning the `widest' acyclic orientations and the independence number

Kempe Chains and Rooted Minors

A (minimal) transversal of a partition is a set which contains exactly one element from each member of the partition and nothing else. A coloring of a graph is a partition of its vertex set into anticliques, that is, sets of pairwise nonadjacent vertices. We study the following problem: Given a transversal $T$ of a proper coloring $\mathfrak{C}$ of some graph $G$, is there a partition $\mathfrak{H}$ of a subset of $V(G)$ into connected sets such that $T$ is a transversal of $\mathfrak{H}$ and such that two sets of $\mathfrak{H}$ are adjacent if their corresponding vertices from $T$ are connected by a path in $G$ using only two colors? It has been conjectured by the first author that for any transversal $T$ of a coloring $\mathfrak{C}$ of order $k$ of some graph $G$ such that any pair of color classes induces a connected graph, there exists such a partition $\mathfrak{H}$ with pairwise adjacent sets (which would prove Hadwiger's conjecture for the class of uniquely optimally colorable graphs); this is open for each $k \geq 5$, here we give a proof for the case that $k=5$ and the subgraph induced by $T$ is connected. Moreover, we show that for $k\geq 7$, it is not sufficient for the existence of $\mathfrak{H}$ as above just to force any two transversal vertices to be connected by a 2-colored path

Breaking Instance-Independent Symmetries In Exact Graph Coloring

Code optimization and high level synthesis can be posed as constraint satisfaction and optimization problems, such as graph coloring used in register allocation. Graph coloring is also used to model more traditional CSPs relevant to AI, such as planning, time-tabling and scheduling. Provably optimal solutions may be desirable for commercial and defense applications. Additionally, for applications such as register allocation and code optimization, naturally-occurring instances of graph coloring are often small and can be solved optimally. A recent wave of improvements in algorithms for Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests generic problem-reduction methods, rather than problem-specific heuristics, because (1) heuristics may be upset by new constraints, (2) heuristics tend to ignore structure, and (3) many relevant problems are provably inapproximable. Problem reductions often lead to highly symmetric SAT instances, and symmetries are known to slow down SAT solvers. In this work, we compare several avenues for symmetry breaking, in particular when certain kinds of symmetry are present in all generated instances. Our focus on reducing CSPs to SAT allows us to leverage recent dramatic improvement in SAT solvers and automatically benefit from future progress. We can use a variety of black-box SAT solvers without modifying their source code because our symmetry-breaking techniques are static, i.e., we detect symmetries and add symmetry breaking predicates (SBPs) during pre-processing. An important result of our work is that among the types of instance-independent SBPs we studied and their combinations, the simplest and least complete constructions are the most effective. Our experiments also clearly indicate that instance-independent symmetries should mostly be processed together with instance-specific symmetries rather than at the specification level, contrary to what has been suggested in the literature