Finding 1-factors in bipartite regular graphs, and edge-coloring bipartite graphs

This paper gives a new and faster algorithm to find a 1-factor in a bipartite D-regular graph. The time complexity of this algorithm is O(n D + n log n log D), where n is the number of nodes. This implies an O(n log n log D + m log D) algorithm to edge-color a bipartite graph with n nodes, m edges and maximum degree D

Vertex-Coloring Edge-Weighting of Bipartite Graphs with Two Edge Weights

Let $G$ be a graph and $\mathcal {S}$ be a subset of $Z$. A vertex-coloring $\mathcal {S}$-edge-weighting of $G$ is an assignment of weight $s$ by the elements of $\mathcal {S}$ to each edge of $G$ so that adjacent vertices have different sums of incident edges weights. It was proved that every 3-connected bipartite graph admits a vertex-coloring $\{1,2\}$-edge-weighting (Lu, Yu and Zhang, (2011) \cite{LYZ}). In this paper, we show that the following result: if a 3-edge-connected bipartite graph $G$ with minimum degree $\delta$ contains a vertex $u\in V(G)$ such that $d_G(u)=\delta$ and $G-u$ is connected, then $G$ admits a vertex-coloring $\mathcal {S}$-edge-weighting for $\mathcal {S}\in \{\{0,1\},\{1,2\}\}$. In particular, we show that every 2-connected and 3-edge-connected bipartite graph admits a vertex-coloring $\mathcal {S}$-edge-weighting for $\mathcal {S}\in \{\{0,1\},\{1,2\}\}$. The bound is sharp, since there exists a family of infinite bipartite graphs which are 2-connected and do not admit vertex-coloring $\{1,2\}$-edge-weightings or vertex-coloring $\{0,1\}$-edge-weightings.Comment: In this paper, we show that every 2-connected and 3-edge-connected bipartite graph admits a vertex-coloring S-edge-weighting for S\in {{0,1},{1,2}

Vertex-Coloring 2-Edge-Weighting of Graphs

A $k$-{\it edge-weighting} $w$ of a graph $G$ is an assignment of an integer weight, $w(e)\in \{1,\dots, k\}$, to each edge $e$. An edge weighting naturally induces a vertex coloring $c$ by defining $c(u)=\sum_{u\sim e} w(e)$ for every $u \in V(G)$. A $k$-edge-weighting of a graph $G$ is \emph{vertex-coloring} if the induced coloring $c$ is proper, i.e., $c(u) \neq c(v)$ for any edge $uv \in E(G)$. Given a graph $G$ and a vertex coloring $c_0$, does there exist an edge-weighting such that the induced vertex coloring is $c_0$? We investigate this problem by considering edge-weightings defined on an abelian group. It was proved that every 3-colorable graph admits a vertex-coloring $3$-edge-weighting \cite{KLT}. Does every 2-colorable graph (i.e., bipartite graphs) admit a vertex-coloring 2-edge-weighting? We obtain several simple sufficient conditions for graphs to be vertex-coloring 2-edge-weighting. In particular, we show that 3-connected bipartite graphs admit vertex-coloring 2-edge-weighting

Some results on the palette index of graphs

Given a proper edge coloring $\varphi$ of a graph $G$, we define the palette $S_{G}(v,\varphi)$ of a vertex $v \in V(G)$ as the set of all colors appearing on edges incident with $v$. The palette index $\check s(G)$ of $G$ is the minimum number of distinct palettes occurring in a proper edge coloring of $G$. In this paper we give various upper and lower bounds on the palette index of $G$ in terms of the vertex degrees of $G$, particularly for the case when $G$ is a bipartite graph with small vertex degrees. Some of our results concern $(a,b)$-biregular graphs; that is, bipartite graphs where all vertices in one part have degree $a$ and all vertices in the other part have degree $b$. We conjecture that if $G$ is $(a,b)$-biregular, then $\check{s}(G)\leq 1+\max\{a,b\}$, and we prove that this conjecture holds for several families of $(a,b)$-biregular graphs. Additionally, we characterize the graphs whose palette index equals the number of vertices

Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

Perfect Matchings as IID Factors on Non-Amenable Groups

We prove that in every bipartite Cayley graph of every non-amenable group, there is a perfect matching that is obtained as a factor of independent uniform random variables. We also discuss expansion properties of factors and improve the Hoffman spectral bound on independence number of finite graphs.Comment: 16 pages; corrected missing reference in v