Neighbor sum distinguishing edge coloring of subcubic graphs

A proper edge-k-coloring of a graph G is a mapping from E(G) to {1, 2,..., k} such that no two adjacent edges receive the same color. A proper edge-k-coloring of G is called neighbor sum distinguishing if for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. Let χ_Σ'(G) denote the smallest value k in such a coloring of G. This parameter makes sense for graphs containing no isolated edges(we call such graphs normal). The maximum average degree mad(G) of G is the maximum of the average degrees of its non-empty subgraphs. In this paper, we prove that if G is a normal subcubic graph with mad(G) 5/2,then χ_Σ'(G) ≤ 5. We also prove that if G is a normal subcubic graph with at least two 2-vertices, 6 colors are enough for a neighbor sum distinguishing edge coloring of G, which holds for the list version as well.     

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.     

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.     

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.     

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}.     

On total colorings of some special 1-planar graphs

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we verify the total coloring conjecture for every 1-planar graph G if either Δ(G) ≥ 9 and g(G) ≥ 4, or Δ(G) ≥ 7 and g(G) ≥ 5, where Δ(G) is the maximum degree of G and g(G) is the girth of G.     

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 \(\delta_{g}\left(G\right) = min_{v\epsilon V (G)}\left\{\lfloor\frac{d\left(v\right)}{g\left(v\right)}\rfloor\right\}\).     

Some infinite families of Ramsey ($${\varvec{P}}_\mathbf{3},{\varvec{P}}_{{\varvec{n}}}$$)-minimal trees

For any given two graphs G and H, the notation \(F\rightarrow \) (G, H) means that for any red–blue coloring of all the edges of F will create either a red subgraph isomorphic to G or a blue subgraph isomorphic to H. A graph F is a Ramsey (G, H)-minimal graph if \(F\rightarrow \) (G, H) but \(F-e\nrightarrow (G,H)\), for every \(e \in E(F)\). The class of all Ramsey (G, H)-minimal graphs is denoted by \(\mathcal {R}(G,H)\). In this paper, we construct some infinite families of trees belonging to \(\mathcal {R}(P_3,P_n)\), for \(n=8\) and 9. In particular, we give an algorithm to obtain an infinite family of trees belonging to \(\mathcal {R}(P_3,P_n)\), for \(n\ge 10\).     

Edge clique covering sum of graphs

The edge clique cover sum number (resp. edge clique partition sum number) of a graph G, denoted by scc(G) (resp. scp(G)), is defined as the smallest integer k for which there exists a collection of complete subgraphs of G, covering (resp. partitioning) all edges of G such that the sum of sizes of the cliques is at most k. By definition, scc(G) \({\leqq}\) scp(G). Also, it is known that for every graph G on n vertices, scp(G) \({\leqq n^{2}/2}\). In this paper, among some other results, we improve this bound for scc(G). In particular, we prove that if G is a graph on n vertices with no isolated vertex and the maximum degree of the complement of G is d ? 1, for some integer d, then scc(G) \({\leqq cnd\left\lceil\log \left(({n-1})/(d-1)\right)\right\rceil}\), where c is a constant. Moreover, we conjecture that this bound is best possible up to a constant factor. Using a well-known result by Bollobás on set systems, we prove that this conjecture is true at least for d = 2. Finally, we give an interpretation of this conjecture as an interesting set system problem which can be viewed as a multipartite generalization of Bollobás’ two families theorem.     

<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.     

A note on the independent domination number versus the domination number in bipartite graphs

Let γ(G) and i(G) be the domination number and the independent domination number of G, respectively. Rad and Volkmann posted a conjecture that i(G)/γ(G) ≤ Δ(G)/2 for any graph G, where Δ(G) is its maximum degree (see N. J. Rad, L. Volkmann (2013)). In this work, we verify the conjecture for bipartite graphs. Several graph classes attaining the extremal bound and graphs containing odd cycles with the ratio larger than Δ(G)/2 are provided as well.     

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.     

Conditional connectivity of bubble sort graphs

A subset F ? V (G) is called an R ^k -vertex-cut of a graph G if G ? F is disconnected and each vertex of G ? F has at least k neighbors in G ? F. The R ^k -vertex-connectivity of G, denoted by κ ^k (G), is the cardinality of a minimum R ^k -vertex-cut of G. Let B _n be the bubble sort graph of dimension n. It is known that κ _k (B _n ) = 2 ^k (n ? k ? 1) for n ≥ 2k and k = 1, 2. In this paper, we prove it for k = 3 and conjecture that it is true for all k ∈ N. We also prove that the connectivity cannot be more than conjectured.     

Neighbor Sum Distinguishing Index

We consider proper edge colorings of a graph G using colors of the set {1, . . . , k}. Such a coloring is called neighbor sum distinguishing if for any pair of adjacent vertices x and y the sum of colors taken on the edges incident to x is different from the sum of colors taken on the edges incident to y. The smallest value of k in such a coloring of G is denoted by ndi_Σ(G). In the paper we conjecture that for any connected graph G ≠ C ₅ of order n ≥ 3 we have ndi_Σ(G) ≤ Δ(G) + 2. We prove this conjecture for several classes of graphs. We also show that ndi_Σ(G) ≤ 7Δ(G)/2 for any graph G with Δ(G) ≥ 2 and ndi_Σ(G) ≤ 8 if G is cubic.     

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.     

Note on a conjecture for the sum of signless Laplacian eigenvalues

For a simple graph G on n vertices and an integer k with 1 ? k ? n, denote by \(\mathcal{S}^+_k\) (G) the sum of k largest signless Laplacian eigenvalues of G. It was conjectured that \(\mathcal{S}^+_k(G)\leqslant{e}(G)+(^{k+1}_{\;\;2})\) (G) ? e(G) + (k+1 2), where e(G) is the number of edges of G. This conjecture has been proved to be true for all graphs when k ∈ {1, 2, n ? 1, n}, and for trees, unicyclic graphs, bicyclic graphs and regular graphs (for all k). In this note, this conjecture is proved to be true for all graphs when k = n ? 2, and for some new classes of graphs.     

On integral Cayley sum graphs

Let S be a subset of a finite abelian group G. The Cayley sum graph Cay⁺(G, S) of G with respect to S is a graph whose vertex set is G and two vertices g and h are joined by an edge if and only if g + h ∈ S. We call a finite abelian group G a Cayley sum integral group if for every subset S of G, Cay⁺(G, S) is integral i.e., all eigenvalues of its adjacency matrix are integers. In this paper, we prove that all Cayley sum integral groups are represented by Z₃ and Zn₂ ⁿ, n ≥ 1, where Z_k is the group of integers modulo k. Also, we classify simple connected cubic integral Cayley sum graphs.     

On <Emphasis Type="Italic">k</Emphasis>- critical 2<Emphasis Type="Italic">k</Emphasis>- connected graphs

A graph G is called an (n,k)-graph if κ(G-S)=n-|S| for any S ? V(G) with |S| ≤ k, where ?(G) denotes the connectivity of G. Mader conjectured that for k ≥ 3 the graph K_2k+2?(1-factor) is the unique (2k, k)-graph. Kriesell has settled two special cases for k = 3,4. We prove the conjecture for the general case k ≥ 5.     

Large $${\varvec{p}}^{\prime }$$-orbits for $${\varvec{p}}$$p-nilpotent linear groups

Let G be a p-nilpotent linear group on a finite vector space V of characteristic p. Suppose that |G||V| is odd. Let P be a Sylow p-subgroup of G. We show that there exist vectors \(v_1\) and \(v_2\) in V such that \(C_G(v_1) \cap C_G(v_2)=P\). A striking conjecture of Malle and Navarro offers a simple global criterion for the nilpotence (in the sense of Broué and Puig) of a p-block of a finite group. Our result implies that this conjecture holds for groups of odd order.     

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 (\(\chi (G)\)).