Neighbor sum distinguishing colorings of graphs with maximum average degree less than $$\tfrac{{37}} {{12}}$$

Let G be a graph and let its maximum degree and maximum average degree be denoted by Δ(G) and mad(G), respectively. A neighbor sum distinguishing k-edge colorings of graph G is a proper k-edge coloring of graph G such that, for any edge uv ∈ E(G), the sum of colors assigned on incident edges of u is different from the sum of colors assigned on incident edges of v. The smallest value of k in such a coloring of G is denoted by χ′_∑(G). Flandrin et al. proposed the following conjecture that χ′_∑ (G) ≤ Δ(G) + 2 for any connected graph with at least 3 vertices and G ≠ C₅. In this paper, we prove that the conjecture holds for a normal graph with mad(G) < \(\tfrac{{37}}{{12}}\) and Δ(G) ≥ 7.     

Acyclic edge coloring of planar graphs without adjacent cycles

A proper edge coloring of a graph G is said to be acyclic if there is no bicolored cycle in G.The acyclic edge chromatic number of G,denoted byχ′a(G),is the smallest number of colors in an acyclic edge coloring of G.Let G be a planar graph with maximum degree.In this paper,we show thatχ′a(G)+2,if G has no adjacent i-and j-cycles for any i,j∈{3,4,5},which implies a result of Hou,Liu and Wu(2012);andχ′a(G)+3,if G has no adjacent i-and j-cycles for any i,j∈{3,4,6}.     

Neighbor sum distinguishing chromatic index of sparse graphs via the combinatorial Nullstellensatz

Let ?: E(G) → {1, 2, · · ·, k} be an edge coloring of a graph G. A proper edge-k-coloring of G is called neighbor sum distinguishing if \(\sum\limits_{e \mathrel\backepsilon u} {\phi \left( e \right)} \ne \sum\limits_{e \mathrel\backepsilon v} {\phi \left( e \right)} \) for each edge uv ∈ E(G). The smallest value k for which G has such a coloring is denoted by χ′_Σ(G), which makes sense for graphs containing no isolated edge (we call such graphs normal). It was conjectured by Flandrin et al. that χ′_Σ(G) ≤ Δ(G) + 2 for all normal graphs, except for C₅. Let mad(G) = \(\max \left\{ {\frac{{2\left| {E\left( h \right)} \right|}}{{\left| {V\left( H \right)} \right|}}|H \subseteq G} \right\}\) be the maximum average degree of G. In this paper, we prove that if G is a normal graph with Δ(G) ≥ 5 and mad(G) < 3 ? \(\frac{2}{{\Delta \left( G \right)}}\), then χ′_Σ(G) ≤ Δ(G) + 1. This improves the previous results and the bound Δ(G) + 1 is sharp.     

Neighbor sum distinguishing total chromatic number of <Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>-minor free graph

A k-total coloring of a graph G is a mapping ?: V (G) ? E(G) → {1; 2,..., k} such that no two adjacent or incident elements in V (G) ? E(G) receive the same color. Let f(v) denote the sum of the color on the vertex v and the colors on all edges incident with v: We say that ? is a k-neighbor sum distinguishing total coloring of G if f(u) 6 ≠ f(v) for each edge uv ∈ E(G): Denote χ _Σ ^″ (G) the smallest value k in such a coloring of G: Pil?niak and Wo?niak conjectured that for any simple graph with maximum degree Δ(G), χ _Σ ^″ ≤ Δ(G)+3. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that for K ₄-minor free graph G with Δ(G) > 5; χ _Σ ^″ = Δ(G) + 1 if G contains no two adjacent Δ-vertices, otherwise, χ _Σ ^″ (G) = Δ(G) + 2.     

Some class 1 graphs on <Emphasis Type="Italic">g</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript>-colorings

An edge-coloring of a graph G is an assignment of colors to all the edges of G. A g _c -coloring of a graph G is an edge-coloring of G such that each color appears at each vertex at least g(v) times. The maximum integer k such that G has a g _c -coloring with k colors is called the g _c -chromatic index of G and denoted by \(\chi\prime_{g_{c}}\)(G). In this paper, we extend a result on edge-covering coloring of Zhang and Liu in 2011, and give a new sufficient condition for a simple graph G to satisfy \(\chi\prime_{g_{c}}\)(G) = δ _g (G), where \(\delta_{g}\left(G\right) = min_{v\epsilon V (G)}\left\{\lfloor\frac{d\left(v\right)}{g\left(v\right)}\rfloor\right\}\).     

Acyclic colorings of graphs with bounded degree

A k-coloring (not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colors i and j the subgraph induced by the edges whose endpoints have colors i and j is acyclic. We consider some generalized acyclic k-colorings, namely, we require that each color class induces an acyclic or bounded degree graph. Mainly we focus on graphs with maximum degree 5. We prove that any such graph has an acyclic 5-coloring such that each color class induces an acyclic graph with maximum degree at most 4. We prove that the problem of deciding whether a graph G has an acyclic 2-coloring in which each color class induces a graph with maximum degree at most 3 is NP-complete, even for graphs with maximum degree 5. We also give a linear-time algorithm for an acyclic t-improper coloring of any graph with maximum degree d assuming that the number of colors is large enough.     

A variation of a conjecture due to Erdös and Sós

Erdoes and Soes conjectured in 1963 that every graph G on n vertices with edge number e（G） 〉 1/2（k - 1）n contains every tree T with k edges as a subgraph. In this paper, we consider a variation of the above conjecture, that is, for n 〉 9/ 2k^2 ＋ 37/2＋ 14 and every graph G on n vertices with e（G） 〉 1/2 （k- 1）n, we prove that there exists a graph G＇ on n vertices having the same degree sequence as G and containing every tree T with k edges as a subgraph.     

Improved bounds on the generalized acyclic chromatic number

An r-acyclic edge chromatic number of a graph G, denoted by a _r ^′ r(G), is the minimum number of colors used to produce an edge coloring of the graph such that adjacent edges receive different colors and every cycle C has at least min {|C|, r} colors. We prove that a _r ^′ r(G) ≤ (4r + 1)Δ(G), when the girth of the graph G equals to max{50, Δ(G)} and 4 ≤ r ≤ 7. If we relax the restriction of the girth to max {220, Δ(G)}, the upper bound of a _r ^′ r(G) is not larger than (2r + 5)Δ(G) with 4 ≤ r ≤ 10.     

A note on the minimum total coloring of planar graphs

Graph coloring is an important tool in the study of optimization,computer science,network design,e.g.,file transferring in a computer network,pattern matching,computation of Hessians matrix and so on.In this paper,we consider one important coloring,vertex coloring of a total graph,which is also called total coloring.We consider a planar graph G with maximum degree Δ(G)≥8,and proved that if G contains no adjacent i,j-cycles with two chords for some i,j∈{5,6,7},then G is total-(Δ+1)-colorable.     

On DP-coloring of graphs and multigraphs

While solving a question on the list coloring of planar graphs, Dvo?ák and Postle introduced the new notion of DP-coloring (they called it correspondence coloring). A DP-coloring of a graph G reduces the problem of finding a coloring of G from a given list L to the problem of finding a “large” independent set in the auxiliary graph H(G,L) with vertex set {(v, c): v ∈ V (G) and c ∈ L(v)}. It is similar to the old reduction by Plesnevi? and Vizing of the k-coloring problem to the problem of finding an independent set of size |V(G)| in the Cartesian product G□K _k, but DP-coloring seems more promising and useful than the Plesnevi?–Vizing reduction. Some properties of the DP-chromatic number χ _DP (G) resemble the properties of the list chromatic number χ _l (G) but some differ quite a lot. It is always the case that χ _DP (G) ≥ χ _l (G). The goal of this note is to introduce DP-colorings for multigraphs and to prove for them an analog of the result of Borodin and Erd?s–Rubin–Taylor characterizing the multigraphs that do not admit DP-colorings from some DP-degree-lists. This characterization yields an analog of Gallai’s Theorem on the minimum number of edges in n-vertex graphs critical with respect to DP-coloring.     

Relaxed Locally Identifying Coloring of Graphs

A locally identifying coloring (lid-coloring) of a graph is a proper vertex-coloring such that the sets of colors appearing in the closed neighborhoods of any pair of adjacent vertices having distinct neighborhoods are distinct. Our goal is to study a relaxed locally identifying coloring (rlid-coloring) of a graph that is similar to locally identifying coloring for which the coloring is not necessarily proper. We denote by \(\chi _{rlid}(G)\) the minimum number of colors used in a relaxed locally identifying coloring of a graph G. In this paper, we prove that the problem of deciding that \(\chi _{rlid}(G)=3\) for a 2-degenerate planar graph G is NP-complete and for a bipartite graph G is polynomial. We give several bounds of \(\chi _{rlid}(G)\) for different families of graphs and construct new graphs for which these bounds are tight. We also compare this parameter with the minimum number of colors used in a locally identifying coloring of a graph G (\(\chi _{lid}(G)\)), the size of a minimum identifying code of G (\(\gamma _{id}(G)\)) and the chromatic number of G (\(\chi (G)\)).     

Proper connection number of bipartite graphs

An edge-colored graph G is proper connected if every pair of vertices is connected by a proper path. The proper connection number of a connected graph G, denoted by pc(G), is the smallest number of colors that are needed to color the edges of G in order to make it proper connected. In this paper, we obtain the sharp upper bound for pc(G) of a general bipartite graph G and a series of extremal graphs. Additionally, we give a proper 2-coloring for a connected bipartite graph G having δ(G) ≥ 2 and a dominating cycle or a dominating complete bipartite subgraph, which implies pc(G) = 2. Furthermore, we get that the proper connection number of connected bipartite graphs with δ ≥ 2 and diam(G) ≤ 4 is two.     

On <Emphasis Type="Italic">g</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript>-colorings of nearly bipartite graphs

Let G be a simple graph, let d(v) denote the degree of a vertex v and let g be a nonnegative integer function on V (G) with 0 ≤ g(v) ≤ d(v) for each vertex v ∈ V (G). A g _c -coloring of G is an edge coloring such that for each vertex v ∈ V (G) and each color c, there are at least g(v) edges colored c incident with v. The g _c -chromatic index of G, denoted by χ′g _c (G), is the maximum number of colors such that a gc-coloring of G exists. Any simple graph G has the g _c -chromatic index equal to δ _g (G) or δ _g (G) ? 1, where \({\delta _g}\left( G \right) = \mathop {\min }\limits_{v \in V\left( G \right)} \left\lfloor {d\left( v \right)/g\left( v \right)} \right\rfloor \). A graph G is nearly bipartite, if G is not bipartite, but there is a vertex u ∈ V (G) such that G ? u is a bipartite graph. We give some new sufficient conditions for a nearly bipartite graph G to have χ′g _c (G) = δ _g (G). Our results generalize some previous results due to Wang et al. in 2006 and Li and Liu in 2011.     

Acyclic edge chromatic number of outerplanar graphs

A proper edge coloring of a graph G is called acyclic if there is no 2‐colored cycle in G. The acyclic edge chromatic number of G, denoted by χ(G), is the least number of colors in an acyclic edge coloring of G. In this paper, we determine completely the acyclic edge chromatic number of outerplanar graphs. The proof is constructive and supplies a polynomial time algorithm to acyclically color the edges of any outerplanar graph G using χ(G) colors. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 22–36, 2010     

List neighbor sum distinguishing edge coloring of subcubic graphs

A proper $k$-edge-coloring of a graph with colors in $\{1,2,\dots ,k\}$ is neighbor sum distinguishing (or, NSD for short) if for any two adjacent vertices, the sums of the colors of the edges incident with each of them are distinct. Flandrin et al. conjectured that every connected graph with at least $6$ vertices has an NSD edge coloring with at most $\Delta +2$ colors. Huo et al. proved that every subcubic graph without isolated edges has an NSD $6$-edge-coloring. In this paper, we first prove a structural result about subcubic graphs by applying the decomposition theorem of Trotignon and Vu?kovi?, and then applying this structural result and the Combinatorial Nullstellensatz, we extend the NSD $6$-edge-coloring result to its list version and show that every subcubic graph without isolated edges has a list NSD $6$-edge-coloring.     

<Emphasis Type="Italic">r</Emphasis>-Dynamic Chromatic Number of Some Line Graphs

An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that |c(N(v))| ≥ min {r, deg(v)}, for each v ∈ V (G). The r-dynamic chromatic number of a graph G is the smallest k such that G admits an r-dynamic coloring with k colors. In this paper, we obtain the r-dynamic chromatic number of the line graph of helm graphs H_n for all r between minimum and maximum degree of H_n. Moreover, our proofs are constructive, what means that we give also polynomial time algorithms for the appropriate coloring. Finally, as the first, we define an equivalent model for edge coloring.     

Chromatic capacity and graph operations

The chromatic capacityχ_cap(G) of a graph G is the largest k for which there exists a k-coloring of the edges of G such that, for every coloring of the vertices of G with the same colors, some edge is colored the same as both its vertices. We prove that there is an unbounded function f:N→N such that χ_cap(G)?f(χ(G)) for almost every graph G, where χ denotes the chromatic number. We show that for any positive integers n and k with k?n/2 there exists a graph G with χ(G)=n and χ_cap(G)=n-k, extending a result of Greene. We obtain bounds on that are tight as r→∞, where is the complete n-partite graph with r vertices in each part. Finally, for any positive integers p and q we construct a graph G with χ_cap(G)+1=χ(G)=p that contains no odd cycles of length less than q.     

Improved upper bounds on acyclic edge colorings

An acyclic edge coloring of a graph is a proper edge coloring such that every cycle contains edges of at least three distinct colors.The acyclic chromatic index of a graph G,denoted by a′(G),is the minimum number k such that there is an acyclic edge coloring using k colors.It is known that a′(G)≤16△for every graph G where △denotes the maximum degree of G.We prove that a′(G)13.8△for an arbitrary graph G.We also reduce the upper bounds of a′(G)to 9.8△and 9△with girth 5 and 7,respectively.     

Injective choosability of subcubic planar graphs with girth 6

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 $\left|L\left(v\right)\right|\ge k$ for all $v\in V\left(G\right)$, $G$ 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 the result of Lu?ar, ?krekovski, and Tancer that subcubic planar graphs with girth at least 7 are injectively 5-colorable. Our result also improves several other results in particular cases.     

<Emphasis Type="Italic">T</Emphasis>-Coloring of Certain Networks

Given a graph G and a finite set T of non-negative integers containing zero, a T-coloring of G is a non-negative integer function f defined on V(G) such that \(|f(x)-f(y)|\not \in T\) whenever \((x,y)\in E(G)\). The span of T-coloring is the difference between the largest and smallest colors, and the T-span of G is the minimum span over all T-colorings f of G. The edge span of a T-coloring is the maximum value of \(|f(x)-f(y)|\) over all edges \((x,y)\in E(G)\), and the T-edge span of G is the minimum edge span over all T-colorings f of G. In this paper, we compute T-span and T-edge span of crown graph, circular ladder and mobius ladder, generalized theta graph, series-parallel graph and wrapped butterfly network.