A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitioned into subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices of V_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focus on complexity aspects of such colorings when l=2,3. More precisely, we prove that, for any fixed integers k,j,g with (k,j) distinct form (0,0) and g >= 3, either every planar graph with girth at least g is (k,j)-colorable or it is NP-complete to determine whether a planar graph with girth at least g is (k,j)-colorable. Also, for any fixed integer k, it is NP-complete to determine whether a planar graph that is either (0,0,0)-colorable or non-(k,k,1)-colorable is (0,0,0)-colorable. Additionally, we exhibit non-(3,1)-colorable planar graphs with girth 5 and non-(2,0)-colorable planar graphs with girth 7.
A cycle is $2$-colored if its edges are properly colored by two distinct colors. A $(d,s)$-edge colorable graph $G$ is a $d$-regular graph that admits a proper $d$-edge coloring in which every edge of $G$ is in at least $s-1$ $2$-colored $4$-cycles. Given a $(d,s)$-edge colorable graph $G$ and a list assigment $L$ of forbidden colors for the edges of $G$ satisfying certain sparsity conditions, we prove that there is a proper $d$-edge coloring of $G$ that avoids $L$, that is, a proper edge coloring $φ$ of $G$ such that $φ(e) \notin L(e)$ for every edge $e$ of $G$.
A tree-coloring of a maximal planar graph is a proper vertex $4$-coloring such that every bichromatic subgraph, induced by this coloring, is a tree. A maximal planar graph $G$ is tree-colorable if $G$ has a tree-coloring. In this article, we prove that a tree-colorable maximal planar graph $G$ with $δ(G)\geq 4$ contains at least four odd-vertices. Moreover, for a tree-colorable maximal planar graph of minimum degree 4 that contains exactly four odd-vertices, we show that the subgraph induced by its four odd-vertices is not a claw and contains no triangles.
We define a perfect coloring of a graph $G$ as a proper coloring of $G$ such that every connected induced subgraph $H$ of $G$ uses exactly $ω(H)$ many colors where $ω(H)$ is the clique number of $H$. A graph is perfectly colorable if it admits a perfect coloring. We show that the class of perfectly colorable graphs is exactly the class of perfect paw-free graphs. It follows that perfectly colorable graphs can be recognized and colored in linear time.
A graph is locally irregular if the degrees of the end-vertices of every edge are distinct. An edge coloring of a graph G is locally irregular if every color induces a locally irregular subgraph of G. A colorable graph G is any graph which admits a locally irregular edge coloring. The locally irregular chromatic index X'irr(G) of a colorable graph G is the smallest number of colors required by a locally irregular edge coloring of G. The Local Irregularity Conjecture claims that all colorable graphs require at most 3 colors for a locally irregular edge coloring. Recently, it has been observed t…
DP-coloring (also known as correspondence coloring) of a simple graph is a generalization of list coloring. It is known that planar graphs without 4-cycles adjacent to triangles are 4-choosable, and planar graphs without 4-cycles are DP-4-colorable. In this paper, we show that planar graphs without 4-cycles adjacent to triangles are DP-4-colorable, which is an extension of the two results above.
In the past various distance based colorings on planar graphs were introduced. We turn our focus to three of them, namely $2$-distance coloring, injective coloring, and exact square coloring. A $2$-distance coloring is a proper coloring of the vertices in which no two vertices at distance $2$ receive the same color, an injective coloring is a coloring of the vertices in which no two vertices with a common neighbor receive the same color, and an exact square coloring is a coloring of the vertices in which no two vertices at distance exactly $2$ receive the same color. We prove that planar graph…
In order to solve a question on list coloring of planar graphs, Dvořák and Postle introduced the concept of so called DP-coloring, thereby extending the concept of list-coloring. DP-coloring was anaylized in detail by Bernshteyn, Kostochka, and Pron for graphs and multigraphs; they characterized DP-degree colorable multigraphs and deduced a Brooks' type result from this. The characterization of the corresponding 'bad' covers was later given by Kim and Ozeki. In this paper, the concept of DP-colorings is extended to hypergraphs having multiple (hyper-)edges. We characterize the DP-degree colora…
A coloring of a matroid is an assignment of colors to the elements of its ground set. We restrict to proper colorings - those for which elements of the same color form an independent set. Seymour proved that a $k$-colorable matroid is also colorable from any lists of size $k$. We generalize this theorem to the case when lists have still fixed sizes, but not necessarily equal. For any fixed size of lists assignment $\ell$, we prove that, if a matroid is colorable from a particular lists of size $\ell$, then it is colorable from any lists of size $\ell$. This gives an explicit necessary and suff…
Let $G$ be a graph, $r \geq t$ integers, and $N \subseteq E(G)$. An $(r,t)$-threshold-coloring of $G$ with respect to $N$ is a mapping $c: V(G) \rightarrow \{0,\ldots,r-1\}$ such that $|c(u)-c(v)| \leq t$ for every $uv \in N$ and $|c(u)-c(v)|>t$ for every $uv \in E(G) \setminus N$. A graph is total threshold colorable if there exist integers $r,t$ such that for every $N \subseteq E(G)$, $G$ admits an $(r,t)$-threshold-coloring with respect to $N$. We show that every prism is total threshold colorable, and that the Petersen graph is total threshold colorable. In contrast to this fact we show th…
DP-coloring (also known as correspondence coloring) is a generalization of list coloring, introduced by Dvořák and Postle in 2017. It is well-known that there are non-4-choosable planar graphs. Much attention has recently been put on sufficient conditions for planar graphs to be DP-$4$-colorable. In particular, for each $k \in \{3, 4, 5, 6\}$, every planar graph without $k$-cycles is DP-$4$-colorable. In this paper, we prove that every planar graph without $7$-cycles and butterflies is DP-$4$-colorable. Our proof can be easily modified to prove other sufficient conditions that forbid clusters …
In this paper uniquely list colorable graphs are studied. A graph G is called to be uniquely k-list colorable if it admits a k-list assignment from which G has a unique list coloring. The minimum k for which G is not uniquely k-list colorable is called the M-number of G. We show that every triangle-free uniquely vertex colorable graph with chromatic number k+1, is uniquely k-list colorable. A bound for the M-number of graphs is given, and using this bound it is shown that every planar graph has M-number at most 4. Also we introduce list criticality in graphs and characterize all 3-list critica…
A two-dimensional \emph{grid} is a set $\Gnm = [n]\times[m]$. A grid $\Gnm$ is \emph{$c$-colorable} if there is a function $χ_{n,m}: \Gnm \to [c]$ such that there are no rectangles with all four corners the same color. We address the following question: for which values of $n$ and $m$ is $\Gnm$ $c$-colorable? This problem can be viewed as a bipartite Ramsey problem and is related to a the Gallai-Witt theorem (also called the multidimensioanl Van Der Waerden's Theorem). We determine (1) \emph{exactly} which grids are 2-colorable, (2) \emph{exactly} which grids are 3-colorable, and (3) \emph{exa…
DP-coloring is a generalization of a list coloring in simple graphs. Many results in list coloring can be generalized in those of DP-coloring. Kim and Ozeki showed that planar graphs without $k$-cycles where $k=3,4,5,$ or $6$ are DP-$4$-colorable. Recently, Kim and Yu extended the result on $3$- and $4$-cycles by showing that planar graphs without triangles adjacent to $4$-cycles are DP-$4$-colorable. Xu and Wu showed that planar graphs without $5$-cycles adjacent simultaneously to $3$-cycles and $4$-cycles are $4$-choosable. In this paper, we extend the result on $5$-cycles and triangles adja…
An odd coloring of a graph $G$ is a proper coloring such that any non-isolated vertex in $G$ has a coloring appears odd times on its neighbors. The odd chromatic number, denoted by $χ_o(G)$, is the minimum number of colors that admits an odd coloring of $G$. Petruševski and Škrekovski in 2021 introduced this notion and proved that if $G$ is planar, then $χ_o(G)\le9$ and conjectured that $χ_o(G)\le5$. More recently, Petr and Portier improved $9$ to $8$. A graph is $1$-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. Cranston, Lafferty and Song showe…
Hadwiger Conjecture has been an open problem for over a half century1,6, which says that there is at most a complete graph Kt but no Kt+1 for every t-colorable graph. A few cases of Hadwiger Conjecture, such as 1, 2, 3, 4, 5, 6-colorable graphs have been completely proved to convince all1-5, but the proofs are tremendously difficult for over the 5-colorable graph6,7. Although the development of graph theory inspires scientists to understand graph coloring deeply, it is still an open problem for over 7-colorable graphs6,7. Therefore, we put forward a brand new chromatic graph configuration and …
We show that every plane graph with maximum face size four whose all faces of size four are vertex-disjoint is cyclically 5-colorable. This answers a question of Albertson whether graphs drawn in the plane with all crossings independent are 5-colorable.
In 1965, Vizing proved that every planar graph $G$ with maximum degree $Δ\geq 8$ is edge $Δ$-colorable. It is also proved that every planar graph $G$ with maximum degree $Δ=7$ is edge $Δ$-colorable by Sanders and Zhao, independently by Zhang. In this paper, we extend the above results by showing that every $K_5$-minor free graph with maximum degree $Δ$ at least seven is edge $Δ$-colorable.
Let $G$ be a graph without 4-cycles and 5-cycles. We show that the problem to determine whether $G$ is $(0,k)$-colorable is NP-complete for each positive integer $k.$ Moreover, we construct non-$(1,k)$-colorable planar graphs without 4-cycles and 5-cycles for each positive integer $k.$ Finally, we prove that $G$ is $(d_1,d_2)$-colorable where $(d_1,d_2)=(4,4), (3,5),$ and $(2,9).$
Let $M_1,M_2,\ldots,M_k$ be a collection of matroids on the same ground set $E$. A coloring $c:E \rightarrow \{1,2,\ldots,k\}$ is called \emph{cooperative} if for every color $j$, the set of elements in color $j$ is independent in $M_j$. We prove that such coloring always exists provided that every matroid $M_j$ is itself $k$-colorable (the set $E$ can be split into at most $k$ independent sets of $M_j$). We derive this fact from a generalization of Seymour's list coloring theorem for matroids, which asserts that every $k$-colorable matroid is $k$-list colorable, too. We also point on some con…
DP-coloring is a generalization of list coloring, which was introduced by Dvořák and Postle [J. Combin. Theory Ser. B 129 (2018) 38--54]. Zhang [Inform. Process. Lett. 113 (9) (2013) 354--356] showed that every planar graph with neither adjacent triangles nor 5-, 6-, 9-cycles is 3-choosable. Liu et al. [Discrete Math. 342 (2019) 178--189] showed that every planar graph without 4-, 5-, 6- and 9-cycles is DP-3-colorable. In this paper, we show that every planar graph with neither adjacent triangles nor 5-, 6-, 9-cycles is DP-3-colorable, which generalizes these results. Yu et al. gave three Bord…
Odd coloring is a proper coloring with an additional restriction that every non-isolated vertex has some color that appears an odd number of times in its neighborhood. The minimum number of colors $k$ that can ensure an odd coloring of a graph $G$ is denoted by $χ_o(G)$. We say $G$ is odd $k$-colorable if $χ_o(G)\le k$. This notion is introduced very recently by Petruševski and Škrekovski, who proved that if $G$ is planar then $ χ_{o}(G) \leq 9 $. A toroidal graph is a graph that can be embedded on a torus. Note that a $K_7$ is a toroidal graph, $χ_{o}(G)\leq7$. Tian and Yin proved that every …
We prove that every triangle-free graph of tree-width t has chromatic number at most ceil((t + 3)/2), and demonstrate that this bound is tight. The argument also establishes a connection between coloring graphs of tree-width t and on-line coloring of graphs of path-width t.
Odd coloring is a proper coloring with an additional restriction that every non-isolated vertex has some color that appears an odd number of times in its neighborhood. The minimum number of colors $k$ that can ensure an odd coloring of a graph $G$ is denoted by $χ_o(G)$. We say $G$ is $k$-colorable if $χ_o(G)\le k$. This notion is introduced very recently by Petruševski and Škrekovski, who proved that if $G$ is planar then $ χ_{o}(G) \leq 9 $. A toroidal graph is a graph that can be embedded on a torus. Note that a $K_7$ is a toroidal graph, $χ_{o}(G)\leq7$. In this paper, we proved that, ever…
Let G be a graph with n vertices and suppose that for each vertex v in G, there exists a list of k colors L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable graph. Recently M. Mahdian and E.S. Mahmoodian characterized uniquely 2-list colorable graphs. Here we state some results which will pave the way in characterization of uniquely k-list colorable graphs. There is a relationship between this concept and defining sets in graph colorings and critical sets in latin squares.
For $1\leq s_1 \le s_2 \le \ldots \le s_k$ and a graph $G$, a packing $(s_1, s_2, \ldots, s_k)$-coloring of $G$ is a partition of $V(G)$ into sets $V_1, V_2, \ldots, V_k$ such that, for each $1\leq i \leq k$, the distance between any two distinct $x,y\in V_i$ is at least $s_i + 1$. The packing chromatic number, $χ_p(G)$, of a graph $G$ is the smallest $k$ such that $G$ has a packing $(1,2, \ldots, k)$-coloring. It is known that there are trees of maximum degree 4 and subcubic graphs $G$ with arbitrarily large $χ_p(G)$. Recently, there was a series of papers on packing $(s_1, s_2, \ldots, s_k)$…
A graph is locally irregular if the degrees of the end-vertices of every edge are distinct. An edge coloring of a graph G is locally irregular if every color induces a locally irregular subgraph of G. A colorable graph G is any graph which admits a locally irregular edge coloring. The locally irregular chromatic index X'irr(G) of a colorable graph G is the smallest number of colors required by a locally irregular edge coloring of G. The Local Irregularity Conjecture claims that all colorable graphs require at most 3 colors for locally irregular edge coloring. Recently, it has been observed tha…
The star chromatic index of a multigraph $G$, denoted $χ'_{s}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star $k$-edge-colorable if $χ'_{s}(G)\le k$. Dvořák, Mohar and Šámal [Star chromatic index, J Graph Theory 72 (2013), 313--326] proved that every subcubic multigraph is star $7$-edge-colorable, and conjectured that every subcubic multigraph should be star $6$-edge-colorable. Kerdjoudj, Kostochka and Raspaud considered the list version of this problem for simple graphs and proved …
In this paper, we show that every $(2P_2,K_4)$-free graph is 4-colorable. The bound is attained by the five-wheel and the complement of the seven-cycle. This answers an open question by Wagon \cite{Wa80} in the 1980s. Our result can also be viewed as a result in the study of the Vizing bound for graph classes. A major open problem in the study of computational complexity of graph coloring is whether coloring can be solved in polynomial time for $(4P_1,C_4)$-free graphs. Lozin and Malyshev \cite{LM17} conjecture that the answer is yes. As an application of our main result, we provide the first …
A graph $G$ is called interval colorable if it has a proper edge coloring with colors $1,2,3,\dots$ such that the colors of the edges incident to every vertex of $G$ form an interval of integers. Not all graphs are interval colorable; in fact, quite few families have been proved to admit interval colorings. In this paper we introduce and investigate a new notion, the interval coloring thickness of a graph $G$, denoted ${θ_{\mathrm{int}}}(G)$, which is the minimum number of interval colorable edge-disjoint subgraphs of $G$ whose union is $G$. Our investigation is motivated by scheduling problem…
A graph $G$ is $(d_1,\ldots,d_k)$-colorable if its vertex set can be partitioned into $k$ sets $V_1,\ldots,V_k$, such that for each $i\in\{1, \ldots, k\}$, the subgraph of $G$ induced by $V_i$ has maximum degree at most $d_i$. The Four Color Theorem states that every planar graph is $(0,0,0,0)$-colorable, and a classical result of Cowen, Cowen, and Woodall shows that every planar graph is $(2,2,2)$-colorable. In this paper, we extend both of these results to graphs on surfaces. Namely, we show that every graph embeddable on a surface of Euler genus $g>0$ is $(0,0,0,9g-4)$-colorable and $(2,2,9…
Given two graphs $F$ and $G$, an $F$-WORM coloring of $G$ is an assignment of colors to its vertices in such a way that no $F$-subgraph of $G$ is monochromatic or rainbow. If $G$ has at least one such coloring, then it is called $F$-WORM colorable and $W^-(G,F)$ denotes the minimum possible number of colors. Here, we consider $F$-WORM colorings with a fixed 2-connected graph $F$ and prove the following three main results: (1) For every natural number $k$, there exists a graph $G$ which is $F$-WORM colorable and $W^-(G,F)=k$; (2) It is NP-complete to decide whether a graph is $F$-WORM colorable…
Star Coloring on Some Subclasses of Chordal Graphs
Facultad de Ciencias, Universidad Nacional Autónoma de México（墨西哥国立自治大学理学院）；Instituto de Matemáticas, Universidad Nacional Autónoma de México（墨西哥国立自治大学数学研究所）；Departamento de Computação, Universidade Federal do Ceará（巴西联邦塞阿拉大学计算机系）
An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable if it has an interval $t$-coloring for some positive integer $t$. The problem of deciding whether a bipartite graph is interval colorable is NP-complete. The smallest known examples of interval non-colorable bipartite graphs have $19$ vertices. On the other hand it is known that the bipartite graphs on at most $14$ vertices are interval colorable. In th…
Neumann-Lara (1985) and Škrekovski conjectured that every planar digraph with digirth at least three is 2-colorable. We prove a relaxed version of this conjecture: every planar digraph of digirth at least five is 2-colorable. The result also holds in the setting of list colorings.
Let $k$ be an positive integer. For any two integers $i$ and $j$ in $\{0,1,\dots,k-1\}$, let $|i-j|_k=\min\{|i-j|,k-|i-j|\}$ be the circular distance between $i$ and $j$. Let $t$ be a nonnegative integer. Suppose $f$ is a mapping from $V(G)$ to $\{0,1,\dots,k-1\}$. If adjacent vertices receive different integers, and for each vertex $u$ of $G$, the number of neighbors $v$ of $u$ with $|f(u)-f(v)|_k=1$ is at most $t$, then $f$ is called a $t$-relaxed 2-distant circular $k$-coloring, or simply a $(\frac{k}{2},t)^*$-coloring of $G$. If $G$ has a $(\frac{k}{2},t)^*$-coloring, then $G$ is called $(…
An edge-coloring of a multigraph $G$ with colors $1,\ldots,t$ is called an interval $t$-coloring if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. In this note, we show that all Eulerian multigraphs with an odd number of edges have no interval coloring. We also give some methods for constructing of interval non-edge-colorable Eulerian multigraphs.
A proper vertex coloring of a graph $G$ is $r$-dynamic if for each $v\in V(G)$, at least $\min\{r,d(v)\}$ colors appear in $N_G(v)$. In this paper we investigate $r$-dynamic versions of coloring, list coloring, and paintability. We prove that planar and toroidal graphs are 3-dynamically 10-colorable, and this bound is sharp for toroidal graphs. We also give bounds on the minimum number of colors needed for any $r$ in terms of the genus of the graph: for sufficiently large $r$, every graph with genus $g$ is $r$-dynamically $((r+1)(g+5)+3)$-colorable when $g\leq2$ and $r$-dynamically $((r+1)(2g+…
We consider the problem of coloring a 3-colorable graph in polynomial time using as few colors as possible. We present a combinatorial algorithm getting down to $\tO(n^{4/11})$ colors. This is the first combinatorial improvement of Blum's $\tO(n^{3/8})$ bound from FOCS'90. Like Blum's algorithm, our new algorithm composes nicely with recent semi-definite approaches. The current best bound is $O(n^{0.2072})$ colors by Chlamtac from FOCS'07. We now bring it down to $O(n^{0.2038})$ colors.
In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\geq 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. We show that every graph with no $K_t$ minor is $O(t(\log t)^β)$-colorable for every $β> 1/4$, making the first improvement on the order of magnitude of the Kostochka-Thomason bound.
Montassier, Raspaud, and Wang (2006) asked to find the smallest positive integers $d_0$ and $d_1$ such that planar graphs without $\{4,5\}$-cycles and $d^Δ\ge d_0$ are $3$-choosable and planar graphs without $\{4,5,6\}$-cycles and $d^Δ\ge d_1$ are $3$-choosable, where $d^Δ$ is the smallest distance between triangles. They showed that $2\le d_0\le 4$ and $d_1\le 3$. In this paper, we show that the following planar graphs are DP-3-colorable: (1) planar graphs without $\{4,5\}$-cycles and $d^Δ\ge 3$ are DP-$3$-colorable, and (2) planar graphs without $\{4,5,6\}$-cycles and $d^Δ\ge 2$ are DP-$3$-c…
Hadwiger's conjecture from 1943 states that for every integer $t\ge1$, every graph either can be $t$-colored or has a subgraph that can be contracted to the complete graph on $t+1$ vertices. As pointed out by Paul Seymour in his recent survey on Hadwiger's conjecture, proving that graphs with no $K_7$ minor are $6$-colorable is the first case of Hadwiger's conjecture that is still open. It is not known yet whether graphs with no $K_7$ minor are $7$-colorable. Using a Kempe-chain argument along with the fact that an induced path on three vertices is dominating in a graph with independence numbe…
A total $k$-coloring of a graph is an assignment of $k$ colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The Total Coloring Conjecture (TCC) states that every simple graph $G$ has a total ($Δ(G)+2$)-coloring, where $Δ(G)$ is the maximum degree of $G$. This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph $G$ of $Δ(G) \geq 9$ or $Δ(G) \in \{7, 8\}$ with some restrictions has a total $(Δ(G) + 1)$-colori…
Given a 3-colorable graph $X$, the 3-coloring complex $B(X)$ is the graph whose vertices are all the independent sets which occur as color classes in some 3-coloring of $X$. Two color classes $C,D \in V(B(X))$ are joined by an edge if $C$ and $D$ appear together in a 3-coloring of $X$. The graph $B(X)$ is 3-colorable. Graphs for which $B(B(X))$ is isomorphic to $X$ are termed reflexive graphs. In this paper, we consider 3-edge-colorings of cubic graphs for which we allow half-edges. Then we consider the 3-coloring complexes of their line graphs. The main result of the paper is a surprising out…
The paper deals with an extremal problem concerning equitable colorings of uniform hyper\-graph. Recall that a vertex coloring of a hypergraph $H$ is called proper if there are no monochro-matic edges under this coloring. A hypergraph is said to be equitably $r$-colorable if there is a proper coloring with $r$ colors such that the sizes of any two color classes differ by at most one. In the present paper we prove that if the number of edges $|E(H)|\leq 0.01\left(\frac{n}{\ln n}\right)^{\frac {r-1}{r}}r^{n-1}$ then the hypergraph $H$ is equitably $r$-colorable provided $r<\sqrt[5]{\ln n}$.
Let $k$ and $r$ be two integers with $k \ge 2$ and $k\ge r \ge 1$. In this paper we show that (1) if a strongly connected digraph $D$ contains no directed cycle of length $1$ modulo $k$, then $D$ is $k$-colorable; and (2) if a digraph $D$ contains no directed cycle of length $r$ modulo $k$, then $D$ can be vertex-colored with $k$ colors so that each color class induces an acyclic subdigraph in $D$. The first result gives an affirmative answer to a question posed by Tuza in 1992, and the second implies the following strong form of a conjecture of Diwan, Kenkre and Vishwanathan: If an undirected…
A locally irregular graph is a graph in which the end-vertices of every edge have distinct degrees. A locally irregular edge coloring of a graph G is any edge coloring of G such that each of the colors induces a locally irregular subgraph of G. A graph G is colorable if it admits a locally irregular edge coloring. The locally irregular chromatic index of a colorable graph G, denoted by CHI'irr(G), is the smallest number of colors used by a locally irregular edge coloring of G. The Local Irregularity Conjecture claims that all graphs, except odd length path, odd length cycle and a certain class…
A graph G is (a:b)-colorable if there exists an assignment of b-element subsets of {1,...,a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We first show that for every triangle-free planar graph G and a vertex x of G, the graph G has a set coloring phi by subsets of {1,...,6} such that |phi(v)|>=2 for each vertex v of G and |phi(x)|=3. As a corollary, every triangle-free planar graph on n vertices is (6n:2n+1)-colorable. We further use this result to prove that for every Delta, there exists a constant M_Delta such that every planar graph G of girth at least five a…
In the first partial result toward Steinberg's now-disproved three coloring conjecture, Abbott and Zhou used a counting argument to show that every planar graph without cycles of lengths 4 through 11 is 3-colorable. Implicit in their proof is a fact about plane graphs: in any plane graph of minimum degree 3, if no two triangles share an edge, then triangles make up strictly less than 2/3 of the faces. We show how this result, combined with Kostochka and Yancey's resolution of Ore's conjecture for k = 4, implies that every planar graph without cycles of lengths 4 through 8 is 3-colorable.
