A node-capacitated Okamura-Seymour theorem

The classical Okamura-Seymour theorem states that for an edge-capacitated, multi-commodity flow instance in which all terminals lie on a single face of a planar graph, there exists a feasible concurrent flow if and only if the cut conditions are satisfied. Simple examples show that a similar theorem is impossible in the node-capacitated setting. Nevertheless, we prove that an approximate flow/cut theorem does hold: For some universal c > 0, if the node cut conditions are satisfied, then one can simultaneously route a c-fraction of all the demands. This answers an open question of Chekuri and Kawarabayashi. More generally, we show that this holds in the setting of multi-commodity polymatroid networks introduced by Chekuri, et. al. Our approach employs a new type of random metric embedding in order to round the convex programs corresponding to these more general flow problems.Comment: 30 pages, 5 figure

Vertex-Coloring with Star-Defects

Defective coloring is a variant of traditional vertex-coloring, according to which adjacent vertices are allowed to have the same color, as long as the monochromatic components induced by the corresponding edges have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to the size, degree, and acyclicity of the monochromatic components. In this paper we focus on defective colorings in which the monochromatic components are acyclic and have small diameter, namely, they form stars. For outerplanar graphs, we give a linear-time algorithm to decide if such a defective coloring exists with two colors and, in the positive case, to construct one. Also, we prove that an outerpath (i.e., an outerplanar graph whose weak-dual is a path) always admits such a two-coloring. Finally, we present NP-completeness results for non-planar and planar graphs of bounded degree for the cases of two and three colors

Colouring exact distance graphs of chordal graphs

For a graph $G=(V,E)$ and positive integer $p$, the exact distance-$p$ graph $G^{[\natural p]}$ is the graph with vertex set $V$ and with an edge between vertices $x$ and $y$ if and only if $x$ and $y$ have distance $p$. Recently, there has been an effort to obtain bounds on the chromatic number $\chi(G^{[\natural p]})$ of exact distance-$p$ graphs for $G$ from certain classes of graphs. In particular, if a graph $G$ has tree-width $t$, it has been shown that $\chi(G^{[\natural p]}) \in \mathcal{O}(p^{t-1})$ for odd $p$, and $\chi(G^{[\natural p]}) \in \mathcal{O}(p^{t}\Delta(G))$ for even $p$. We show that if $G$ is chordal and has tree-width $t$, then $\chi(G^{[\natural p]}) \in \mathcal{O}(p\, t^2)$ for odd $p$, and $\chi(G^{[\natural p]}) \in \mathcal{O}(p\, t^2 \Delta(G))$ for even $p$. If we could show that for every graph $H$ of tree-width $t$ there is a chordal graph $G$ of tree-width $t$ which contains $H$ as an isometric subgraph (i.e., a distance preserving subgraph), then our results would extend to all graphs of tree-width $t$. While we cannot do this, we show that for every graph $H$ of genus $g$ there is a graph $G$ which is a triangulation of genus $g$ and contains $H$ as an isometric subgraph.Comment: 11 pages, 2 figures. Versions 2 and 3 include minor changes, which arise from reviewers' comment

Crossing Minimization for 1-page and 2-page Drawings of Graphs with Bounded Treewidth

We investigate crossing minimization for 1-page and 2-page book drawings. We show that computing the 1-page crossing number is fixed-parameter tractable with respect to the number of crossings, that testing 2-page planarity is fixed-parameter tractable with respect to treewidth, and that computing the 2-page crossing number is fixed-parameter tractable with respect to the sum of the number of crossings and the treewidth of the input graph. We prove these results via Courcelle's theorem on the fixed-parameter tractability of properties expressible in monadic second order logic for graphs of bounded treewidth.Comment: Graph Drawing 201

To Prove Four Color Theorem

In this paper, we give a proof for four color theorem(four color conjecture). Our proof does not involve computer assistance and the most important is that it can be generalized to prove Hadwiger Conjecture. Moreover, we give algorithms to color and test planarity of planar graphs, which can be generalized to graphs containing $K_x(x>5)$ minor. There are four parts of this paper: Part-1: To Prove Four Color Theorem Part-2: An Equivalent Statement of Hadwiger Conjecture when $k=5$ Part-3: A New Proof of Wagner's Equivalence Theorem Part-4: A Geometric View of Outerplanar GraphComment: The paper is further reduced, and each part is more self-contained, is the fina