A proper $[k]$-total coloring $c$ of a graph $G$ is a mapping $c$ from $V(G)\bigcup E(G)$ to $[k]=\{1,2,\cdots,k\}$ such that $c(x)\neq c(y)$ for which $x$, $y\in V(G)\bigcup E(G)$ and $x$ is adjacent to or incident with $y$. Let $\prod(v)$ denote the product of $c(v)$ and the colors on all the edges incident with $v$. For each edge $uv\in E(G)$, if $\prod(u)\neq \prod(v)$, then the coloring $c$ is called a neighbor product distinguishing total coloring of $G$. we use $χ"_{\prod}(G)$ to denote the minimal value of $k$ in such a coloring of $G$. In 2015, Li et al. conjectured that $Δ(G)+3$ colors enable a graph to have a neighbor product distinguishing total coloring. In this paper, we consider the neighbor product distinguishing total coloring of corona product $G\circ H$, and obtain that $χ"_{\prod}(G\circ H)\leq Δ(G\circ H)+3$.
{\emph A star edge-coloring} of a graph is a proper edge-coloring without bichromatic paths and cycles of length four. In this paper, we consider the list version of this coloring and prove that the list star chromatic index of every subcubic graph is at most $7$, answering the question of Dvořák et al. (Star chromatic index, J. Graph Theory 72 (2013), 313--326).
A strong edge coloring of a graph $G$ is a proper edge coloring in which each color class is an induced matching of $G$. In 1993, Brualdi and Quinn Massey proposed a conjecture that every bipartite graph without $4$-cycles and with the maximum degrees of the two partite sets $2$ and $Δ$ admits a strong edge coloring with at most $Δ+2$ colors. We prove that this conjecture holds for such graphs with $Δ=3$. We also confirm the conjecture proposed by Faudree et al. for subcubic bipartite graphs.
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 $(s_1, s_2, \ldots, s_k)$…
Given a graph $G$ and a nondecreasing sequence $S=(s_1,\ldots,s_k)$ of positive integers, the mapping $c:V(G)\longrightarrow \{1,\ldots,k\}$ is called an $S$-packing coloring of $G$ if for any two distinct vertices $x$ and $y$ in $c^{-1}(i)$, the distance between $x$ and $y$ is greater than $s_i$. The smallest integer $k$ such that there exists a $(1,2,\ldots,k)$-packing coloring of a graph $G$ is called the packing chromatic number of $G$, denoted $χ_ρ(G)$. The question of boundedness of the packing chromatic number in the class of subcubic (planar) graphs was investigated in several earlier …
Given an edge-coloring of a simple graph, assign to every vertex $v$ a set $S_v$ comprised of the colors used on the edges incident to $v$. The $k$-intersection chromatic index of a graph is the minimum $t$ such that the edge set can be properly $t$-colored, additionally requiring that for every two adjacent vertices $u$ and $v$, $|S_u \cap S_v| \le k$. For all $k \neq 2$, this value is known for subcubic planar graphs, and furthermore, these values are best possible. We naturally extend this definition to multigraphs with bounded edge multiplicity, and we show that every subcubic planar multi…
A $k$-coloring of a graph $G=(V,E)$ is called semi-equitable if there exists a partition of its vertex set into independent subsets $V_1,\ldots,V_k$ in such a way that $|V_1| \notin \{\lceil |V|/k\rceil, \lfloor |V|/k \rfloor\}$ and $||V_i|-|V_j|| \leq 1$ for each $i,j=2,\ldots,k$. The color class $V_1$ is called non-equitable. In this note we consider the complexity of semi-equitable $k$-coloring, $k\geq 4$, of the vertices of a cubic or subcubic graph $G$. In particular, we show that, given a $n$-vertex subcubic graph $G$ and constants $ε> 0$, $k \geq 4$, it is NP-complete to obtain a semi-e…
For a sequence of non-decreasing positive integers $S = (s_1, \ldots, s_k)$, a packing $S$-coloring is a partition of $V(G)$ into sets $V_1, \ldots, V_k$ 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 smallest $k$ such that $G$ has a packing $(1,2, \ldots, k)$-coloring is called the packing chromatic number of $G$ and is denoted by $χ_p(G)$. For a graph $G$, let $D(G)$ denote the graph obtained from $G$ by subdividing every edge. The question whether $χ_p(D(G)) \le 5$ for all subcubic graphs was first asked by Gastineau and Togni…
On $(1,2^4)$ and $(1,2^5)$-packing edge-coloring of sparse subcubic graphs
Konkuk University（建国大学）；Korea Institute for Advanced Study (KIAS)（韩国高等研究院）；Xi’an Jiaotong-Liverpool University（西交利物浦大学）；Guizhou Minzu University（贵州民族大学）
A strong $k$-edge-coloring of a graph G is an edge-coloring with $k$ colors in which every color class is an induced matching. The strong chromatic index of $G$, denoted by $χ'_{s}(G)$, is the minimum $k$ for which $G$ has a strong $k$-edge-coloring. In 1985, Erdős and Nešetřil conjectured that $χ'_{s}(G)\leq\frac{5}{4}Δ(G)^2$, where $Δ(G)$ is the maximum degree of $G$. When $G$ is a graph with maximum degree at most 3, the conjecture was verified independently by Andersen and Horák, Qing, and Trotter. In this paper, we consider the list version of strong edge-coloring. In particular, we show …
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 …
A star edge coloring of a graph is a proper edge coloring with no $2$-colored path or cycle of length four. The star chromatic index $χ'_{st}(G)$ of $G$ is the minimum number $t$ for which $G$ has a star edge coloring with $t$ colors. We prove upper bounds for the star chromatic index of complete bipartite graphs; in particular we obtain tight upper bounds for the case when one part has size at most $3$. We also consider bipartite graphs $G$ where all vertices in one part have maximum degree $2$ and all vertices in the other part has maximum degree $b$. Let $k$ be an integer ($k\geq 1$), we pr…
Thomassen formulated the following conjecture: Every $3$-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree at most $1$ (that is, it consists of a matching and some isolated vertices) and the red subgraph has minimum degree at least $1$ and contains no $3$-edge path. Since all monochromatic components are small in this coloring and there is a certain irregularity, we call such a coloring \emph{crumby}. Recently, Bellitto, Klimošová, Merker, Witkowski and Yuditsky \cite{counter} constructed an infinite family refuting the above conjecture. Their …
An injective edge-coloring $c$ of a graph $G$ is an edge-coloring such that if $e_1$, $e_2$, and $e_3$ are three consecutive edges in $G$ (they are consecutive if they form a path or a cycle of length three), then $e_1$ and $e_3$ receive different colors. The minimum integer $k$ such that, $G$ has an injective edge-coloring with $k$ colors, is called the injective chromatic index of $G$ ($χ'_{\textrm{inj}}(G)$). This parameter was introduced by Cardoso et \textit{al.} \cite{CCCD} motivated by the Packet Radio Network problem. They proved that computing $χ'_{\textrm{inj}}(G)$ of a graph $G$ is …
An injective coloring of a graph $G$ is an assignment of colors to the vertices of $G$ so that any two vertices with a common neighbor have distinct colors. A graph $G$ is injectively $k$-choosable if for any list assignment $L$, where $|L(v)| \geq k$ for all $v \in V(G)$, $G$ has an injective $L$-coloring. Injective colorings have applications in the theory of error-correcting codes and are closely related to other notions of colorability. In this paper, we show that subcubic planar graphs with girth at least 6 are injectively 5-choosable. This strengthens a result of Lužar, Škrekovski, and T…
A graph is \textit{locally irregular} if the neighbors of every vertex $v$ have degrees distinct from the degree of $v$. \textit{locally irregular edge-coloring} of a graph $G$ is an (improper) edge-coloring such that the graph induced on the edges of any color class is locally irregular. It is conjectured that $3$ colors suffice for a locally irregular edge-coloring. Recently, Bensmail et al. (Bensmail, Merker, Thomassen: Decomposing graphs into a constant number of locally irregular subgraphs, {\em European J. Combin.}, 60:124--134, 2017) settled the first constant upper bound for the proble…
We study the exact square chromatic number of subcubic planar graphs. An exact square coloring of a graph G is a vertex-coloring in which any two vertices at distance exactly 2 receive distinct colors. The smallest number of colors used in such a coloring of G is its exact square chromatic number, denoted $χ^{\sharp 2}(G)$. This notion is related to other types of distance-based colorings, as well as to injective coloring. Indeed, for triangle-free graphs, exact square coloring and injective coloring coincide. We prove tight bounds on special subclasses of planar graphs: subcubic bipartite pla…
Impact of local girth on the S-packing coloring of k-saturated subcubic graphs
University of Sciences and Arts in Lebanon（黎巴嫩科学与艺术学院）；KALMA Laboratory, Faculty of Sciences, Lebanese University（黎巴嫩大学理学院 KALMA 实验室）；Al Maaref University（阿尔马雷夫大学）；Lebanese International University (LIU)（黎巴嫩国际大学）
Let $G$ be a graph and $k$ a positive integer. A strong $k$-edge-coloring of $G$ is a mapping $ϕ: E(G)\to \{1,2,\dots,k\}$ such that for any two edges $e$ and $e'$ that are either adjacent to each other or adjacent to a common edge, $ϕ(e)\neq ϕ(e')$. The strong chromatic index of $G$, denoted as $χ'_{s}(G)$, is the minimum integer $k$ such that $G$ has a strong $k$-edge-coloring. Lv, Li and Zhang [Graphs and Combinatorics 38 (3) (2022) 63] proved that if $G$ is a claw-free subcubic graph other than the triangular prism then $χ_s'(G)\le 8$. In addition, they asked if the upper bound $8$ can be …
On (1,1,2,3)- and (1,1,3,3,3)-Packing Colorings of Claw-Free Subcubic Graphs
Lebanese University（黎巴嫩大学）；Lebanese International University (LIU)（黎巴嫩国际大学）；Al Maaref University（Al Maaref大学）；University of Sciences and Arts in Lebanon（黎巴嫩科学与艺术大学）；Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800（波尔多大学、法国国家科学研究中心、波尔多理工学院、LaBRI实验室）
A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induce a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and every assignment of lists of size four to the vertices of the graph, there exists a linear coloring such that the color of each vertex belongs to the list assigned to that vertex and the neighbors of every degree-two vertex receive different colors, unless the graph is $C_5$ or $K_{3,3}$. This confirms a conjecture raised by Esperet, Montassier, and Raspaud…
For a non-decreasing sequence $S = (s_1, \ldots, s_k)$ of positive integers, an $S$-packing edge-coloring of a graph $G$ is a decomposition of edges of $G$ into disjoint sets $E_1, \ldots, E_k$ such that for each $1 \le i \le k$ the distance between any two distinct edges $e_1, e_2 \in E_i$ is at least $s_i+1$. The notion of $S$-packing edge-coloring was first generalized by Gastineau and Togni from its vertex counterpart. They showed that there are subcubic graphs that are not $(1,2,2,2,2,2,2)$-packing (abbreviated to $(1,2^6)$-packing) edge-colorable and asked the question whether every subc…
LaBRI, CNRS, University of Bordeaux（拉布里实验室，法国国家科学研究中心，波尔多大学）；Department of Computer Science, Faculty of Mathematics, Physics and Informatics, Comenius University（布拉迪斯拉发康斯坦丁神学院数学物理与信息学院计算机科学系）
A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $χ_p(G)$, of a graph $G$ is the minimum $k$ such that $G$ has a packing $k$-coloring. Sloper showed that there are $4$-regular graphs with arbitrarily large packing chromatic number. The question whether the packing chromatic number of subcubic graphs is bounded appears in several papers. We answer this question in the negative. Moreover, we show that for every fixed $k$ a…
In a proper edge-coloring the edges of every color form a matching. A matching is induced if the end-vertices of its edges induce a matching. A strong edge-coloring is an edge-coloring in which the edges of every color form an induced matching. We consider intermediate types of edge-colorings, where edges of some colors are allowed to form matchings, and the remaining form induced matchings. Our research is motivated by the conjecture proposed in a recent paper of Gastineau and Togni on S-packing edge-colorings (On S-packing edge-colorings of cubic graphs, Discrete Appl. Math. 259 (2019), 63-7…
\textit{A star edge coloring} of a graph is a proper edge coloring without bichromatic paths and cycles of length four. In this paper we establish tight upper bounds for trees and subcubic outerplanar graphs, and derive an upper bound for outerplanar graphs.
Using computational techniques we provide a framework for proving results on subclasses of planar graphs via discharging method. The aim of this paper is to apply these techniques to study the 2-distance coloring of planar subcubic graphs. Applying these techniques we show that every subcubic planar graph $G$ of girth at least 8 has 2-distance chromatic number at most 6.
Thomassen formulated the following conjecture: Every $3$-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree $1$ (that is, it consists of a matching and some isolated vertices) and the red subgraph has minimum degree at least $1$ and contains no $3$-edge path. We prove the conjecture for Generalized Petersen graphs. We indicate that a coloring with the same properties might exist for any subcubic graph. We confirm this statement for all subcubic trees.
The {\em square} $G^2$ of a graph $G$ is the graph with the same vertex set as $G$ and with two vertices adjacent if their distance in $G$ is at most 2. Thomassen showed that every planar graph $G$ with maximum degree $Δ(G)=3$ satisfies $χ(G^2)\leq 7$. Kostochka and Woodall conjectured that for every graph, the list-chromatic number of $G^2$ equals the chromatic number of $G^2$, that is $χ_l(G^2)=χ(G^2)$ for all $G$. If true, this conjecture (together with Thomassen's result) implies that every planar graph $G$ with $Δ(G)=3$ satisfies $χ_l(G^2)\leq 7$. We prove that every connected graph (not …
An $(a:b)$-coloring of a graph $G$ is a function $f$ which maps the vertices of $G$ into $b$-element subsets of some set of size $a$ in such a way that $f(u)$ is disjoint from $f(v)$ for every two adjacent vertices $u$ and $v$ in $G$. The fractional chromatic number $χ_f(G)$ is the infimum of $a/b$ over all pairs of positive integers $a,b$ such that $G$ has an $(a:b)$-coloring. Heckman and Thomas conjectured that the fractional chromatic number of every triangle-free graph $G$ of maximum degree at most three is at most 2.8. Hatami and Zhu proved that $χ_f(G) \leq 3-3/64 \approx 2.953$. Lu and …
Given a non-decreasing sequence $S=(s\_1,s\_2, \ldots, s\_k)$ of positive integers, an {\em $S$-packing coloring} of a graph $G$ is a mapping $c$ from $V(G)$ to $\{s\_1,s\_2, \ldots, s\_k\}$ such that any two vertices with color $s\_i$ are at mutual distance greater than $s\_i$, $1\le i\le k$. This paper studies $S$-packing colorings of (sub)cubic graphs. We prove that subcubic graphs are $(1,2,2,2,2,2,2)$-packing colorable and $(1,1,2,2,3)$-packing colorable. For subdivisions of subcubic graphs we derive sharper bounds, and we provide an example of a cubic graph of order $38$ which is not $(1…
An injective $k$-edge-coloring of a graph $G$ is an assignment of colors, i.e. integers in $\{1, \ldots , k\}$, to the edges of $G$ such that any two edges each incident with one distinct endpoint of a third edge, receive distinct colors. The problem of determining whether such a $k$-coloring exists is called k-INJECTIVE EDGE-COLORING. We show that 3-INJECTIVE EDGE-COLORING is NP-complete, even for triangle-free cubic graphs, planar subcubic graphs of arbitrarily large girth, and planar bipartite subcubic graphs of girth~6. 4-INJECTIVE EDGE-COLORING remains NP-complete for cubic graphs. For an…
A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $χ_p(G)$, of a graph $G$ is the minimum $k$ such that $G$ has a packing $k$-coloring. For a graph $G$, let $D(G)$ denote the graph obtained from $G$ by subdividing every edge. The questions on the value of the maximum of $χ_p(G)$ and of $χ_p(D(G))$ over the class of subcubic graphs $G$ appear in several papers. Gastineau and Togni asked whether $χ_p(D(G))\leq 5$ for any su…
A proper edge coloring of a graph is strong if it creates no bichromatic path of length three. It is well known that for a strong edge coloring of a $k$-regular graph at least $2k-1$ colors are needed. We show that a $k$-regular graph admits a strong edge coloring with $2k-1$ colors if and only if it covers the Kneser graph $K(2k-1,k-1)$. In particular, a cubic graph is strongly $5$-edge-colorable whenever it covers the Petersen graph. One of the implications of this result is that a conjecture about strong edge colorings of subcubic graphs proposed by Faudree et al. [Ars Combin. 29 B (1990), …
A facial unique-maximum coloring of a plane graph is a vertex coloring where on each face $α$ the maximal color appears exactly once on the vertices of $α$. If the coloring is required to be proper, then the upper bound for the minimal number of colors required for such a coloring is set to $5$. Fabrici and Göring [Fabrici and Goring 2016] even conjectured that $4$ colors always suffice. Confirming the conjecture would hence give a considerable strengthening of the Four Color Theorem. In this paper, we prove that the conjecture holds for subcubic plane graphs, outerplane graphs and plane quadr…
Dominator coloring of a graph is a proper (vertex) coloring with the property that every vertex is either alone in its color class or adjacent to all vertices of at least one color class. A dominated coloring of a graph is a proper coloring such that every color class is dominated with at least one vertex. The dominator chromatic number of corona products and of edge corona products is determined. Sharp lower and upper bounds are given for the dominated chromatic number of edge corona products. The dominator chromatic number of hierarchical products is bounded from above and the dominated chro…
An adjacent vertex distinguishing total $k$-coloring $f$ of a graph $G$ is a proper total $k$-coloring of $G$ such that no pair of adjacent vertices has the same color sets, where the color set at a vertex $v$, $C^G_f(v)$, is $\{f(v)\} \cup \{f(vu)|u \in V (G), vu \in E(G)\}$. In 2005 Zhang et al. posted the conjecture (AVDTCC) that every simple graph $G$ has adjacent vertex distinguishing total $(Δ(G)+3)$-coloring. In this paper we confirm the conjecture for many coronas, in particular for generalized, simple and $l$-coronas of graphs, not relating the results to particular graph classes.
A facial unique-maximum coloring of a plane graph is a proper coloring of the vertices using positive integers such that each face has a unique vertex that receives the maximum color in that face. Fabrici and Göring (2016) proposed a strengthening of the Four Color Theorem conjecturing that all plane graphs have a facial unique-maximum coloring using four colors. This conjecture has been disproven for general plane graphs and it was shown that five colors suffice. In this paper we show that plane graphs, where vertices of degree at least four induce a star forest, are facially unique-maximum 4…
A vertex coloring of a graph $G$ is an assignment of colors to the vertices of $G$ such that every two adjacent vertices of $G$ have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the vertices of a graph, in which the process of labeling is done according to an extra condition. A set $S$ of vertices of a graph $G$ is a dominating set in $G$ if every vertex outside of $S$ is adjacent to at least one vertex belonging to $S$. A domination parameter of $G$ is related to those structures of a graph satisfying some domination property together …
An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycle s. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic e dge coloring using k colors and is denoted by $a'(G)$. A graph is called 2-$degenerate$ if any of its induced subgraph has a vertex of degree at most 2. The class of 2-$degenerate graphs$ properly conta in $series$-$parallel graphs$, $outerplanar graphs$, \emph{non-regular subcubic graphs}, \emph{planar graphs of girth at least 6} and \emph{circle graphs of girth at least 5} as subclasse…
