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

List version of (pp,1)-total labellings

The ($p$,1)-total number $\lambda_p^T(G)$ of a graph $G$ is the width of the smallest range of integers that suffices to label the vertices and the edges of $G$ such that no two adjacent vertices have the same label, no two incident edges have the same label and the difference between the labels of a vertex and its incident edges is at least $p$. In this paper we consider the list version. Let $L(x)$ be a list of possible colors for all $x\in V(G)\cup E(G)$. Define $C_{p,1}^T(G)$ to be the smallest integer $k$ such that for every list assignment with $|L(x)|=k$ for all $x\in V(G)\cup E(G)$, $G$ has a ($p$,1)-total labelling $c$ such that $c(x)\in L(x)$ for all $x\in V(G)\cup E(G)$. We call $C_{p,1}^T(G)$ the ($p$,1)-total labelling choosability and $G$ is list $L$-($p$,1)-total labelable. In this paper, we present a conjecture on the upper bound of $C_{p,1}^T$. Furthermore, we study this parameter for paths and trees in Section 2. We also prove that $C_{p,1}^T(K_{1,n})\leq n+2p-1$ for star $K_{1,n}$ with $p\geq2, n\geq3$ in Section 3 and $C_{p,1}^T(G)\leq \Delta+2p-1$ for outerplanar graph with $\Delta\geq p+3$ in Section 4.Comment: 11 pages, 2 figure

Ramsey-nice families of graphs

For a finite family $\mathcal{F}$ of fixed graphs let $R_k(\mathcal{F})$ be the smallest integer $n$ for which every $k$-coloring of the edges of the complete graph $K_n$ yields a monochromatic copy of some $F\in\mathcal{F}$. We say that $\mathcal{F}$ is $k$-nice if for every graph $G$ with $\chi(G)=R_k(\mathcal{F})$ and for every $k$-coloring of $E(G)$ there exists a monochromatic copy of some $F\in\mathcal{F}$. It is easy to see that if $\mathcal{F}$ contains no forest, then it is not $k$-nice for any $k$. It seems plausible to conjecture that a (weak) converse holds, namely, for any finite family of graphs $\mathcal{F}$ that contains at least one forest, and for all $k\geq k_0(\mathcal{F})$ (or at least for infinitely many values of $k$), $\mathcal{F}$ is $k$-nice. We prove several (modest) results in support of this conjecture, showing, in particular, that it holds for each of the three families consisting of two connected graphs with 3 edges each and observing that it holds for any family $\mathcal{F}$ containing a forest with at most 2 edges. We also study some related problems and disprove a conjecture by Aharoni, Charbit and Howard regarding the size of matchings in regular 3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure

JGraphT -- A Java library for graph data structures and algorithms

Mathematical software and graph-theoretical algorithmic packages to efficiently model, analyze and query graphs are crucial in an era where large-scale spatial, societal and economic network data are abundantly available. One such package is JGraphT, a programming library which contains very efficient and generic graph data-structures along with a large collection of state-of-the-art algorithms. The library is written in Java with stability, interoperability and performance in mind. A distinctive feature of this library is the ability to model vertices and edges as arbitrary objects, thereby permitting natural representations of many common networks including transportation, social and biological networks. Besides classic graph algorithms such as shortest-paths and spanning-tree algorithms, the library contains numerous advanced algorithms: graph and subgraph isomorphism; matching and flow problems; approximation algorithms for NP-hard problems such as independent set and TSP; and several more exotic algorithms such as Berge graph detection. Due to its versatility and generic design, JGraphT is currently used in large-scale commercial, non-commercial and academic research projects. In this work we describe in detail the design and underlying structure of the library, and discuss its most important features and algorithms. A computational study is conducted to evaluate the performance of JGraphT versus a number of similar libraries. Experiments on a large number of graphs over a variety of popular algorithms show that JGraphT is highly competitive with other established libraries such as NetworkX or the BGL.Comment: Major Revisio

Message passing for the coloring problem: Gallager meets Alon and Kahale

Message passing algorithms are popular in many combinatorial optimization problems. For example, experimental results show that {\em survey propagation} (a certain message passing algorithm) is effective in finding proper $k$-colorings of random graphs in the near-threshold regime. In 1962 Gallager introduced the concept of Low Density Parity Check (LDPC) codes, and suggested a simple decoding algorithm based on message passing. In 1994 Alon and Kahale exhibited a coloring algorithm and proved its usefulness for finding a $k$-coloring of graphs drawn from a certain planted-solution distribution over $k$-colorable graphs. In this work we show an interpretation of Alon and Kahale's coloring algorithm in light of Gallager's decoding algorithm, thus showing a connection between the two problems - coloring and decoding. This also provides a rigorous evidence for the usefulness of the message passing paradigm for the graph coloring problem. Our techniques can be applied to several other combinatorial optimization problems and networking-related issues.Comment: 11 page

Radio Resource Allocation for Device-to-Device Underlay Communication Using Hypergraph Theory

Device-to-Device (D2D) communication has been recognized as a promising technique to offload the traffic for the evolved Node B (eNB). However, the D2D transmission as an underlay causes severe interference to both the cellular and other D2D links, which imposes a great technical challenge to radio resource allocation. Conventional graph based resource allocation methods typically consider the interference between two user equipments (UEs), but they cannot model the interference from multiple UEs to completely characterize the interference. In this paper, we study channel allocation using hypergraph theory to coordinate the interference between D2D pairs and cellular UEs, where an arbitrary number of D2D pairs are allowed to share the uplink channels with the cellular UEs. Hypergraph coloring is used to model the cumulative interference from multiple D2D pairs, and thus, eliminate the mutual interference. Simulation results show that the system capacity is significantly improved using the proposed hypergraph method in comparison to the conventional graph based one.Comment: 27 pages,10 figure