Conflict-Free Coloring of Planar Graphs

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number chi_CF(G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N[v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N(v), for which vertex v is not a member of its neighborhood. For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. We also give a complete characterization of the computational complexity of conflict-free coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G, but polynomial for outerplanar graphs. Furthermore, deciding whether chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for outerplanar graphs. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs. For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general} planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on Discrete Mathematics) of extended abstract that appears in Proceeedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pp. 1951-196

On edge-group choosability of graphs

In this paper, we study the concept of edge-group choosability of graphs. We say that G is edge k-group choosable if its line graph is k-group choosable. An edge-group choosability version of Vizing conjecture is given. The evidence of our claim are graphs with maximum degree less than 4, planar graphs with maximum degree at least 11, planar graphs without small cycles, outerplanar graphs and near-outerplanar graphs

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

Clique-Relaxed Graph Coloring

We define a generalization of the chromatic number of a graph G called the k-clique-relaxed chromatic number, denoted χ(k)(G). We prove bounds on χ(k)(G) for all graphs G, including corollaries for outerplanar and planar graphs. We also define the k-clique-relaxed game chromatic number, χg(k)(G), of a graph G. We prove χg(2)(G)≤ 4 for all outerplanar graphs G, and give an example of an outerplanar graph H with χg(2)(H) ≥ 3. Finally, we prove that if H is a member of a particular subclass of outerplanar graphs, then χg(2)(H) ≤ 3

The edge chromatic number of outer-1-planar graphs

A graph is outer-1-planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once. In this paper, we completely determine the edge chromatic number of outer 1-planar graphs

The Relaxed Game Chromatic Index of \u3cem\u3ek\u3c/em\u3e-Degenerate Graphs

The (r, d)-relaxed coloring game is a two-player game played on the vertex set of a graph G. We consider a natural analogue to this game on the edge set of G called the (r, d)-relaxed edge-coloring game. We consider this game on trees and more generally, on k-degenerate graphs. We show that if G is k-degenerate with ∆(G) = ∆, then the first player, Alice, has a winning strategy for this game with r = ∆+k−1 and d≥2k2 + 4k