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Register a new userIncidence coloring of graphs with high maximum average degree
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Added by
Herv\\'e Hocquard
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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An incidence of an undirected graph G is a pair $(v,e)$ where $v$ is a vertex of $G$ and $e$ an edge of $G$ incident with $v$. Two incidences $(v,e)$ and $(w,f)$ are adjacent if one of the following holds: (i) $v = w$, (ii) $e = f$ or (iii) $vw = e$ or $f$. An incidence coloring of $G$ assigns a color to each incidence of $G$ in such a way that adjacent incidences get distinct colors. In 2005, Hosseini Dolama emph{et al.}~citep{ds05} proved that every graph with maximum average degree strictly less than $3$ can be incidence colored with $Delta+3$ colors. Recently, Bonamy emph{et al.}~citep{Bonamy} proved that every graph with maximum degree at least $4$ and with maximum average degree strictly less than $frac{7}{3}$ admits an incidence $(Delta+1)$-coloring. In this paper we give bounds for the number of colors needed to color graphs having maximum average degrees bounded by different values between $4$ and $6$. In particular we prove that every graph with maximum degree at least $7$ and with maximum average degree less than $4$ admits an incidence $(Delta+3)$-coloring. This result implies that every triangle-free planar graph with maximum degree at least $7$ is incidence $(Delta+3)$-colorable. We also prove that every graph with maximum average degree less than 6 admits an incidence $(Delta + 7)$-coloring. More generally, we prove that $Delta+k-1$ colors are enough when the maximum average degree is less than $k$ and the maximum degree is sufficiently large.
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For a graph $G$ and integer $qgeq 2$, an edge $q$-coloring of $G$ is an assignment of colors to edges of $G$, such that edges incident on a vertex span at most $q$ distinct colors. The maximum edge $q$-coloring problem seeks to maximize the number of colors in an edge $q$-coloring of a graph $G$. The problem has been studied in combinatorics in the context of {em anti-Ramsey} numbers. Algorithmically, the problem is NP-Hard for $qgeq 2$ and assuming the unique games conjecture, it cannot be approximated in polynomial time to a factor less than $1+1/q$. The case $q=2$, is particularly relevant in practice, and has been well studied from the view point of approximation algorithms. A $2$-factor algorithm is known for general graphs, and recently a $5/3$-factor approximation bound was shown for graphs with perfect matching. The algorithm (which we refer to as the matching based algorithm) is as follows: Find a maximum matching $M$ of $G$. Give distinct colors to the edges of $M$. Let $C_1,C_2,ldots, C_t$ be the connected components that results when M is removed from G. To all edges of $C_i$ give the $(|M|+i)$th color. In this paper, we first show that the approximation guarantee of the matching based algorithm is $(1 + frac {2} {delta})$ for graphs with perfect matching and minimum degree $delta$. For $delta ge 4$, this is better than the $frac {5} {3}$ approximation guarantee proved in {AAAP}. For triangle free graphs with perfect matching, we prove that the approximation factor is $(1 + frac {1}{delta - 1})$, which is better than $5/3$ for $delta ge 3$.
If the vertices of a graph $G$ are colored with $k$ colors such that no adjacent vertices receive the same color and the sizes of any two color classes differ by at most one, then $G$ is said to be equitably $k$-colorable. Let $|G|$ denote the number of vertices of $G$ and $Delta=Delta(G)$ the maximum degree of a vertex in $G$. We prove that a graph $G$ of order at least 6 is equitably $Delta$-colorable if $G$ satisfies $(|G|+1)/3 leq Delta < |G|/2$ and none of its components is a $K_{Delta +1}$.
A strong edge-coloring of a graph $G$ is an edge-coloring such that any two edges on a path of length three receive distinct colors. We denote the strong chromatic index by $chi_{s}(G)$ which is the minimum number of colors that allow a strong edge-coloring of $G$. ErdH{o}s and Nev{s}etv{r}il conjectured in 1985 that the upper bound of $chi_{s}(G)$ is $frac{5}{4}Delta^{2}$ when $Delta$ is even and $frac{1}{4}(5Delta^{2}-2Delta +1)$ when $Delta$ is odd, where $Delta$ is the maximum degree of $G$. The conjecture is proved right when $Deltaleq3$. The best known upper bound for $Delta=4$ is 22 due to Cranston previously. In this paper we extend the result of Cranston to list strong edge-coloring, that is to say, we prove that when $Delta=4$ the upper bound of list strong chromatic index is 22.
We consider acyclic r-colorings in graphs and digraphs: they color the vertices in r colors, each of which induces an acyclic graph or digraph. (This includes the dichromatic number of a digraph, and the arboricity of a graph.) For any girth and sufficiently high degree, we prove the NP-completeness of acyclic r-colorings; our method also implies the known analogue for classical colorings. The proofs use high girth graphs with high arboricity and dichromatic numbers. High girth graphs and digraphs with high chromatic and dichromatic numbers have been well studied; we re-derive the results from a general result about relational systems, which also implies the similar fact about high girth and high arboricity used in the proofs. These facts concern graphs and digraphs of high girth and low degree; we contrast them by considering acyclic colorings of tournaments (which have low girth and high degree). We prove that even though acyclic two-colorability of tournaments is known to be NP-complete, random acyclically r-colorable tournaments allow recovering an acyclic r-coloring in deterministic linear time, with high probablity.
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 vs 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.
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