2-Reducible Two Paths and an Edge Constructing a Path in (2k + 1)-Edge-Connected Graphs

Let k ≥ 5 be an odd integer and G = (V(G), E(G)) be a k-edge-connected graph. For ${X\subseteq V(G),e(X)}$ denotes the number of edges between X and V(G) ? X. We here prove that if ${\{s_i,t_i\}\subseteq X_i\subseteq V(G)(i=1,2),f}$ is an edge between s ₁ and ${s_2,X_1\cap X_2=\emptyset,e(X_1)\le 2k-3,e(X_2)\le 2k-2}$ , and e(Y) ≥ k + 1 for each ${Y\subseteq V(G)}$ with ${Y\cap\{s_1,t_1,s_2,t_2\}=\{s_1,t_2\}}$ , then there exist paths P ₁ and P ₂ such that P _i joins s _i and ${t_i,V(P_i)\subseteq X_i}$ (i = 1, 2) and ${G-f-E(P_1\cup P_2)}$ is (k ? 2)-edge-connected, and in fact we give a generalization of this result.     

Minimal decompositions of graphs into mutually isomorphic subgraphs

SupposeG _n={G ₁, ...,G _k } is a collection of graphs, all havingn vertices ande edges. By aU-decomposition ofG _n we mean a set of partitions of the edge setsE(G _t ) of theG _i , sayE(G _t )== \(\sum\limits_{j = 1}^r {E_{ij} } \) E _ij , such that for eachj, all theE _ij , 1≦i≦k, are isomorphic as graphs. Define the functionU(G _n) to be the least possible value ofr aU-decomposition ofG _n can have. Finally, letU _k (n) denote the largest possible valueU(G) can assume whereG ranges over all sets ofk graphs havingn vertices and the same (unspecified) number of edges. In an earlier paper, the authors showed that $$U_2 (n) = \frac{2}{3}n + o(n).$$ In this paper, the value ofU _k (n) is investigated fork>2. It turns out rather unexpectedly that the leading term ofU _k (n) does not depend onk. In particular we show $$U_k (n) = \frac{3}{4}n + o_k (n),k \geqq 3.$$      

The Signed k-Submatchings in Graphs

A signed k-submatching of a graph G is a function f : E(G) → {?1,1} satisfying f (E _G (v)) ≤ 1 for at least k vertices ${v \in V(G)}$ . The maximum of the values of f (E(G)), taken over all signed k-submatchings f, is called the signed k-submatching number and is denoted by ${\beta_S^{k}(G)}$ . In this paper, sharp bounds on ${\beta_S^{k}(G)}$ for general graphs are presented. Exact values of ${\beta_S^{k}(G)}$ for several classes of graphs are found.     

Connectivity of Strong Products of Graphs

The strong product ${G\boxtimes H}$ of graphs G = (V ₁, E ₁) and H = (V ₂, E ₂) is the graph with vertex set ${V(G \boxtimes H)=V_1\times V_2}$ , where two distinct vertices ${(x_1, x_2), (y_1, y_2)\in V_1\times V_2}$ are adjacent in ${G\boxtimes H}$ if and only if x _i  = y _i or ${x_i y_i\in E_i}$ for i = 1, 2. We introduce so called I-sets and L-sets in the strong product ${G\boxtimes H}$ and prove that every minimum separating set in ${G\boxtimes H}$ is either an I-set or an L-set in ${G\boxtimes H}$ . Some bounds and exact results for connectivity of strong products follow from this characterization. The result is then generalized to an arbitrary number of factors in the strong product.     

On Spanning Disjoint Paths in Line Graphs

Spanning connectivity of graphs has been intensively investigated in the study of interconnection networks (Hsu and Lin, Graph Theory and Interconnection Networks, 2009). For a graph G and an integer s > 0 and for ${u, v \in V(G)}$ with u ≠ v, an (s; u, v)-path-system of G is a subgraph H consisting of s internally disjoint (u,v)-paths. A graph G is spanning s-connected if for any ${u, v \in V(G)}$ with u ≠ v, G has a spanning (s; u, v)-path-system. The spanning connectivity κ*(G) of a graph G is the largest integer s such that G has a spanning (k; u, v)-path-system, for any integer k with 1 ≤ k ≤ s, and for any ${u, v \in V(G)}$ with u ≠ v. An edge counter-part of κ*(G), defined as the supereulerian width of a graph G, has been investigated in Chen et al. (Supereulerian graphs with width s and s-collapsible graphs, 2012). In Catlin and Lai (Graph Theory, Combinatorics, and Applications, vol. 1, pp. 207–222, 1991) proved that if a graph G has 2 edge-disjoint spanning trees, and if L(G) is the line graph of G, then κ*(L(G)) ≥ 2 if and only if κ(L(G)) ≥ 3. In this paper, we extend this result and prove that for any integer k ≥ 2, if G ₀, the core of G, has k edge-disjoint spanning trees, then κ*(L(G)) ≥ k if and only if κ(L(G)) ≥ max{3, k}.     

A special f-edge cover-coloring of multigraphs

An edge cover-coloring of G is called a special (f,g)-edge cover-coloring, if each color appears at each vertex at least f(v) times and the number of multiple edges receive the same color is at most g(vw) for vw∈E(G). Let $\chi_{f_{g}}''$ denote the maximum positive integer k for which using k colors a special (f,g)-edge cover-coloring of G exists. The existence of $\chi_{f_{g}}''$ is discussed and the lower bound of $\chi_{f_{g}}''$ is obtained.     

Degree Sum Conditions for Cyclability in Bipartite Graphs

We denote by G[X, Y] a bipartite graph G with partite sets X and Y. Let d _G (v) be the degree of a vertex v in a graph G. For G[X, Y] and ${S \subseteq V(G),}$ we define ${\sigma_{1,1}(S):=\min\{d_G(x)+d_G(y) : (x,y) \in (X \cap S,Y) \cup (X, Y \cap S), xy \not\in E(G)\}}$ . Amar et al. (Opusc. Math. 29:345–364, 2009) obtained σ _1,1(S) condition for cyclability of balanced bipartite graphs. In this paper, we generalize the result as it includes the case of unbalanced bipartite graphs: if G[X, Y] is a 2-connected bipartite graph with |X| ≥ |Y| and ${S \subseteq V(G)}$ such that σ _1,1(S) ≥ |X| + 1, then either there exists a cycle containing S or ${|S \cap X| > |Y|}$ and there exists a cycle containing Y. This degree sum condition is sharp.     

Rainbow Connection Number, Bridges and Radius

Let G be a connected graph. The notion of rainbow connection number rc(G) of a graph G was introduced by Chartrand et al. (Math Bohem 133:85–98, 2008). Basavaraju et al. (arXiv:1011.0620v1 [math.CO], 2010) proved that for every bridgeless graph G with radius r, ${rc(G)\leq r(r+2)}$ and the bound is tight. In this paper, we show that for a connected graph G with radius r and center vertex u, if we let D ^r  = {u}, then G has r?1 connected dominating sets ${ D^{r-1}, D^{r-2},\ldots, D^{1}}$ such that ${D^{r} \subset D^{r-1} \subset D^{r-2} \cdots\subset D^{1} \subset D^{0}=V(G)}$ and ${rc(G)\leq \sum_{i=1}^{r} \max \{2i+1,b_i\}}$ , where b _i is the number of bridges in E[D ⁱ , N(D ⁱ )] for ${1\leq i \leq r}$ . From the result, we can get that if ${b_i\leq 2i+1}$ for all ${1\leq i\leq r}$ , then ${rc(G)\leq \sum_{i=1}^{r}(2i+1)= r(r+2)}$ ; if b _i  > 2i + 1 for all ${1\leq i\leq r}$ , then ${rc(G)= \sum_{i=1}^{r}b_i}$ , the number of bridges of G. This generalizes the result of Basavaraju et al. In addition, an example is given to show that there exist infinitely graphs with bridges whose rc(G) is only dependent on the radius of G, and another example is given to show that there exist infinitely graphs with bridges whose rc(G) is only dependent on the number of bridges in G.     

The Signed Edge-Domatic Number of a Graph

For a nonempty graph G = (V, E), a signed edge-domination of G is a function ${f: E(G) \to \{1,-1\}}$ such that ${\sum_{e'\in N_{G}[e]}{f(e')} \geq 1}$ for each ${e \in E(G)}$ . The signed edge-domatic number of G is the largest integer d for which there is a set {f ₁,f ₂, . . . , f _d } of signed edge-dominations of G such that ${\sum_{i=1}^{d}{f_i(e)} \leq 1}$ for every ${e \in E(G)}$ . This paper gives an original study on this concept and determines exact values for some special classes of graphs, such as paths, cycles, stars, fans, grids, and complete graphs with even order.     

Smallest Set-Transversals of k-Partitions

We asymptotically solve an open problem raised independently by Sterboul (Colloq Math Soc J Bolyai 3:1387–1404, 1973), Arocha et al. (J Graph Theory 16:319–326, 1992) and Voloshin (Australas J Combin 11:25–45, 1995). For integers n ≥ k ≥ 2, let f(n, k) denote the minimum cardinality of a family ${\mathcal H}$ of k-element sets over an n-element underlying set X such that every partition ${X_1\cup\cdots\cup X_k=X}$ into k nonempty classes completely partitions some ${H\in\mathcal H}$ ;  that is, ${|H\cap X_i|=1}$ holds for all 1 ≤ i ≤ k. This very natural function—whose defining property for k = 2 just means that ${\mathcal H}$ is a connected graph—turns out to be related to several extensively studied areas in combinatorics and graph theory. We prove general estimates from which ${ f(n,k) = (1+o(1))\, \tfrac{2}{n}\,{n\choose k}}$ follows for every fixed k, and also for all k = o(n ^1/3), as n → ∞. Further, we disprove a conjecture of Arocha et al. (1992). The exact determination of f(n,k) for all n and k appears to be far beyond reach to our present knowledge, since e.g. the equality ${f(n,n-2)={n-2\choose 2}-{\rm ex}(n,\{C_3,C_4\})}$ holds, where the last term is the Turán number for graphs of girth 5.     

Structural Properties and Hamiltonicity of Neighborhood Graphs

Let G = (V, E) be a simple undirected graph. N(G) =  (V, E _N ) is the neighborhood graph of the graph G, if and only if $$E_N = \{\{a,b\}\,|\, a \neq b\,\wedge\,\exists\, x \, \in V: \{x,a\} \in E \, \wedge \, \{x,b\} \in E \}.$$ We present several structural properties of N(G) and characterize the hamiltonicity of N(G) by means of chords of a hamiltonian cycle in G.     

Nowhere-Zero 5-Flows and Even (1,2)-Factors

A graph G = (V, E) admits a nowhere-zero k-flow if there exists an orientation H = (V, A) of G and an integer flow ${\varphi:A \to \mathbb{Z}}$ such that for all ${a \in A, 0 < |\varphi(a)| < k}$ . Tutte conjectured that every bridgeless graphs admits a nowhere-zero 5-flow. A (1,2)-factor of G is a set ${F \subseteq E}$ such that the degree of any vertex v in the subgraph induced by F is 1 or 2. Let us call an edge of G, F-balanced if either it belongs to F or both its ends have the same degree in F. Call a cycle of G F-even if it has an even number of F-balanced edges. A (1,2)-factor F of G is even if each cycle of G is F-even. The main result of the paper is that a cubic graph G admits a nowhere-zero 5-flow if and only if G has an even (1,2)-factor.     

Signed k-independence in graphs

Let k ≥ 2 be an integer. A function f: V(G) → {?1, 1} defined on the vertex set V(G) of a graph G is a signed k-independence function if the sum of its function values over any closed neighborhood is at most k ? 1. That is, Σ _x∈N[v] f(x) ≤ k ? 1 for every v ∈ V(G), where N[v] consists of v and every vertex adjacent to v. The weight of a signed k-independence function f is w(f) = Σ _v∈V(G) f(v). The maximum weight w(f), taken over all signed k-independence functions f on G, is the signed k-independence number α _s ^k (G) of G. In this work, we mainly present upper bounds on α _s ^k (G), as for example α _s ^k (G) ≤ n ? 2?(Δ(G) + 2 ? k)/2?, and we prove the Nordhaus-Gaddum type inequality $\alpha _S^k \left( G \right) + \alpha _S^k \left( {\bar G} \right) \leqslant n + 2k - 3$ , where n is the order, Δ(G) the maximum degree and $\bar G$ the complement of the graph G. Some of our results imply well-known bounds on the signed 2-independence number.     

Metrization of Weighted Graphs

We find a set of necessary and sufficient conditions under which the weight ${w: E \rightarrow \mathbb{R}^{+}}$ on the graph G = (V, E) can be extended to a pseudometric ${d : V \times V \rightarrow \mathbb{R}^{+}}$ . We describe the structure of graphs G for which the set ${\mathfrak{M}_{w}}$ of all such extensions contains a metric whenever w is strictly positive. Ordering ${\mathfrak{M}_{w}}$ by the pointwise order, we have found that the posets $({\mathfrak{M}_{w}, \leqslant)}$ contain the least elements ρ _0,w if and only if G is a complete k-partite graph with ${k \, \geqslant \, 2}$ . In this case the symmetric functions ${f : V \times V \rightarrow \mathbb{R}^{+}}$ , lying between ρ _0,w and the shortest-path pseudometric, belong to ${\mathfrak{M}_{w}}$ for every metrizable w if and only if the cardinality of all parts in the partition of V is at most two.     

Binary Labeling of Graphs

Let G = (V, E) be a graph. A mapping f: E(G) → {0, l} ^m is called a mod 2 coding of G, if the induced mapping g: V(G) → {0, l} ^m , defined as \(g(v) = \sum\limits_{u \in V,uv \in E} {f(uv)}\) , assigns different vectors to the vertices of G. Note that all summations are mod 2. Let m(G) be the smallest number m for which a mod 2 coding of G is possible. Trivially, m(G) ≥ ?Log₂ |V|?. Recently, Aigner and Triesch proved that m(G) ≤ ?Log₂ |V|? + 4. In this paper, we determine m(G). More specifically, we prove that if each component of G has at least three vertices, then $$mG = \left\{ {\begin{array}{*{20}c} {k,} & {if \left| V \right| \ne 2^k - 2} \\ {k + 1,} & {else} \\ \end{array} ,} \right.$$ where k = ?Log₂ |V|?.     

On super edge-magic decomposable graphs

Let G be any graph and let {H _i } _i??I be a family of graphs such that $E\left( {H_i } \right) \cap E\left( {H_j } \right) = \not 0$ when i ?? j, ?? _i??I E(H _i ) = E(G) and $E\left( {H_i } \right) \ne \not 0$ for all i ?? I. In this paper we introduce the concept of {H _i } _i??I -super edge-magic decomposable graphs and {H _i } _i??I -super edge-magic labelings. We say that G is {H _i } _i??I -super edge-magic decomposable if there is a bijection ??: V(G) ?? {1,2,..., |V(G)|} such that for each i ?? I the subgraph H _i meets the following two requirements: ??(V(H _i )) = {1,2,..., |V(H _i )|} and {??(a) +??(b): ab ?? E(H _i )} is a set of consecutive integers. Such function ?? is called an {H _i } _i??I -super edge-magic labeling of G. We characterize the set of cycles C _n which are {H ₁, H ₂}-super edge-magic decomposable when both, H ₁ and H ₂ are isomorphic to (n/2)K ₂. New lines of research are also suggested.     

The Monotone Catenary Degree of Krull Monoids

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. The monotone catenary degree c _mon (H) of H is the smallest integer m with the following property: for each ${a \in H}$ and each two factorizations z, z′ of a with length |z| ≤  |z′|, there exist factorizations z = z ₀, ... ,z _k  = z′ of a with increasing lengths—that is, |z ₀| ≤  ... ≤  |z _k |—such that, for each ${i \in [1,k]}$ , z _i arises from z _i-1 by replacing at most m atoms from z _i-1 by at most m new atoms. Up to now there was only an abstract finiteness result for c _mon (H), but the present paper offers the first explicit upper and lower bounds for c _mon (H) in terms of the group invariants of G.     

A Note on the Roman Bondage Number of Planar Graphs

A Roman dominating function on a graph G = (V(G), E(G)) is a labelling ${f : V(G)\rightarrow \{0,1,2\}}$ satisfying the condition that every vertex with label 0 has at least a neighbour with label 2. The Roman domination number γ _R (G) of G is the minimum of ${\sum_{v \in V(G)}{f(v)}}$ over all such functions. The Roman bondage number b _R (G) of G is the minimum cardinality of all sets ${E\subseteq E(G)}$ for which γ _R (G \ E) > γ _R (G). Recently, it was proved that for every planar graph P, b _R (P) ≤ Δ(P) + 6, where Δ(P) is the maximum degree of P. We show that the Roman bondage number of every planar graph does not exceed 15 and construct infinitely many planar graphs with Roman bondage number equal to 7.     

Upper bounds on the signed (k, k)-domatic number

Let G be a graph with vertex set V(G), and let f : V(G) → {?1, 1} be a two-valued function. If k ≥ 1 is an integer and ${\sum_{x\in N[v]} f(x) \ge k}$ for each ${v \in V(G)}$ , where N[v] is the closed neighborhood of v, then f is a signed k-dominating function on G. A set {f ₁,f ₂, . . . ,f _d } of distinct signed k-dominating functions on G with the property that ${\sum_{i=1}^d f_i(x) \le k}$ for each ${x \in V(G)}$ , is called a signed (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed (k, k)-dominating family on G is the signed (k, k)-domatic number of G. In this article we mainly present upper bounds on the signed (k, k)-domatic number, in particular for regular graphs.     

Simultaneous Domination in Graphs

Let \({F_1, F_2, \ldots, F_k}\) be graphs with the same vertex set V. A subset \({S \subseteq V}\) is a simultaneous dominating set if for every i, 1 ≤ i ≤ k, every vertex of F _i not in S is adjacent to a vertex in S in F _i ; that is, the set S is simultaneously a dominating set in each graph F _i . The cardinality of a smallest such set is the simultaneous domination number. We present general upper bounds on the simultaneous domination number. We investigate bounds in special cases, including the cases when the factors, F _i , are r-regular or the disjoint union of copies of K _r . Further we study the case when each factor is a cycle.