Super cyclically edge-connected vertex-transitive graphs of girth at least 5

A cyclic edge-cut of a graph G is an edge set, the removal of which separates two cycles. If G has a cyclic edge-cut, then it is called cyclically separable. We call a cyclically separable graph super cyclically edge-connected, in short, super-λc, if the removal of any minimum cyclic edge-cut results in a component which is a shortest cycle. In [Zhang, Z., Wang, B.: Super cyclically edge-connected transitive graphs. J. Combin. Optim., 22, 549-562 (2011)], it is proved that a connected vertex-transitive graph is super-λc if G has minimum degree at least 4 and girth at least 6, and the authors also presented a class of nonsuper-λc graphs which have degree 4 and girth 5. In this paper, a characterization of k (k≥4)-regular vertex-transitive nonsuper-λc graphs of girth 5 is given. Using this, we classify all k (k≥4)-regular nonsuper-λc Cayley graphs of girth 5, and construct the first infinite family of nonsuper-λc vertex-transitive non-Cayley graphs.     

Colouring games on outerplanar graphs and trees

Let f be a graph function which assigns to each graph H a non-negative integer f(H)≤|V(H)|. The f-game chromatic number of a graph G is defined through a two-person game. Let X be a set of colours. Two players, Alice and Bob, take turns colouring the vertices of G with colours from X. A partial colouring c of G is legal (with respect to graph function f) if for any subgraph H of G, the sum of the number of colours used in H and the number of uncoloured vertices of H is at least f(H). Both Alice and Bob must colour legally (i.e., the partial colouring produced needs to be legal). The game ends if either all the vertices are coloured or there are uncoloured vertices with no legal colour. In the former case, Alice wins the game. In the latter case, Bob wins the game. The f-game chromatic number of G, χ_g(f,G), is the least number of colours that the colour set X needs to contain so that Alice has a winning strategy. Let be the graph function defined as , for any n≥3 and otherwise. Then is called the acyclic game chromatic number of G. In this paper, we prove that any outerplanar graph G has acyclic game chromatic number at most 7. For any integer k, let ?_k be the graph function defined as ?_k(K₂)=2 and ?_k(P_k)=3 (P_k is the path on k vertices) and ?_k(H)=0 otherwise. This paper proves that if k≥8 then for any tree T, χ_g(?_k,T)≤9. On the other hand, if k≤6, then for any integer n, there is a tree T such that χ_g(?_k,T)≥n.     

Extra edge connectivity and isoperimetric edge connectivity

An edge set S of a connected graph G is a k-extra edge cut, if G-S is no longer connected, and each component of G-S has at least k vertices. The cardinality of a minimum k-extra edge cut, denoted by λ_k(G), is the k-extra edge connectivity of G. The kth isoperimetric edge connectivity γ_k(G) is defined as , where ω(U) is the number of edges with one end in U and the other end in . Write β_k(G)=min{ω(U):U⊂V(G),|U|=k}. A graph G with is said to be γ_k-optimal.In this paper, we first prove that λ_k(G)=γ_k(G) if G is a regular graph with girth g?k/2. Then, we show that except for K_3,3 and K₄, a 3-regular vertex/edge transitive graph is γ_k-optimal if and only if its girth is at least k+2. Finally, we prove that a connected d-regular edge-transitive graph with d?6e_k(G)/k is γ_k-optimal, where e_k(G) is the maximum number of edges in a subgraph of G with order k.     

具有两个同阶轨道的双轨道图的圈边连通度

一个边割被称为圈边割,如果该边割能分离图的两个不同圈.如果一个图有圈边割,称该图为圈边可分离的.一个圈边可分离图G的最小圈边割的阶数被称为圈边连通度,记作cλ（G）.定义：ζ（G）=min{w（X）｜X导出G的最短圈},其中w（X）为端点分别在X和V（G）-X中的边的数目.如果一个圈边可分离图G使得cλ（G）=ζ（G）成立,称该图是圈边最优的.Tian和Meng在文章[11]以及Yang et al在文章[15]中研究了两种不同的双轨道图的圈边最优性.本文我们将研究具有两个同阶轨道的双轨道图的圈边连通度.     

Degree conditions for restricted-edge-connectivity and isoperimetric-edge-connectivity to be optimal

For a connected graph G=(V,E), an edge set S⊂E is a k-restricted-edge-cut, if G-S is disconnected and every component of G-S has at least k vertices. The k-restricted-edge-connectivity of G, denoted by λ_k(G), is defined as the cardinality of a minimum k-restricted-edge-cut. The k-isoperimetric-edge-connectivity is defined as , where is the set of edges with one end in U and the other end in . In this note, we give some degree conditions for a graph to have optimal λ_k and/or γ_k.     

The homology of the cyclic coloring complex of simple graphs

Let G be a simple graph on n vertices, and let χG(λ) denote the chromatic polynomial of G. In this paper, we define the cyclic coloring complex, Δ(G), and determine the dimensions of its homology groups for simple graphs. In particular, we show that if G has r connected components, the dimension of (n−3)rd homology group of Δ(G) is equal to (n−(r+1)) plus , where is the rth derivative of χG(λ). We also define a complex ΔC(G), whose r-faces consist of all ordered set partitions [B₁,…,Br+2] where none of the Bi contain an edge of G and where 1∈B₁. We compute the dimensions of the homology groups of this complex, and as a result, obtain the dimensions of the multilinear parts of the cyclic homology groups of C[x₁,…,xn]/{xixj|ij is an edge of G}. We show that when G is a connected graph, the homology of ΔC(G) has nonzero homology only in dimension n−2, and the dimension of this homology group is . In this case, we provide a bijection between a set of homology representatives of ΔC(G) and the acyclic orientations of G with a unique source at v, a vertex of G.     

Spanning trails containing given edges

A graph G is Eulerian-connected if for any u and v in V(G), G has a spanning (u,v)-trail. A graph G is edge-Eulerian-connected if for any e^′ and e^″ in E(G), G has a spanning (e^′,e^″)-trail. For an integer r?0, a graph is called r-Eulerian-connected if for any X⊆E(G) with |X|?r, and for any , G has a spanning (u,v)-trail T such that X⊆E(T). The r-edge-Eulerian-connectivity of a graph can be defined similarly. Let θ(r) be the minimum value of k such that every k-edge-connected graph is r-Eulerian-connected. Catlin proved that θ(0)=4. We shall show that θ(r)=4 for 0?r?2, and θ(r)=r+1 for r?3. Results on r-edge-Eulerian connectivity are also discussed.     

Adaptable choosability of planar graphs with sparse short cycles

Given a (possibly improper) edge colouring F of a graph G, a vertex colouring of G is adapted toF if no colour appears at the same time on an edge and on its two endpoints. A graph G is called (for some positive integer k) if for any list assignment L to the vertices of G, with |L(v)|≥k for all v, and any edge colouring F of G, G admits a colouring c adapted to F where c(v)∈L(v) for all v. This paper proves that a planar graph G is adaptably 3-choosable if any two triangles in G have distance at least 2 and no triangle is adjacent to a 4-cycle.     

Edge-connectivity and edge-disjoint spanning trees

Given a graph G, for an integer c∈{2,…,|V(G)|}, define λ_c(G)=min{|X|:X⊆E(G),ω(G−X)≥c}. For a graph G and for an integer c=1,2,…,|V(G)|−1, define,

     

On the total vertex irregularity strength of trees

A vertex irregular total k-labelling λ:V(G)∪E(G)?{1,2,…,k} of a graph G is a labelling of vertices and edges of G done in such a way that for any different vertices x and y, their weights wt(x) and wt(y) are distinct. The weight wt(x) of a vertex x is the sum of the label of x and the labels of all edges incident with x. The minimum k for which a graph G has a vertex irregular total k-labelling is called the total vertex irregularity strength of G, denoted by . In this paper, we determine the total vertex irregularity strength of trees.     

Acyclic edge colouring of planar graphs without short cycles

Let G=(V,E) be any finite graph. A mapping C:E→[k] is called an acyclic edgek-colouring of G, if any two adjacent edges have different colours and there are no bichromatic cycles in G. In other words, for every pair of distinct colours i and j, the subgraph induced in G by all the edges which have colour i or j, is acyclic. The smallest number k of colours, such that G has an acyclic edge k-colouring is called the acyclic chromatic index of G, denoted by .In 2001, Alon et al. conjectured that for any graph G it holds that ; here Δ(G) stands for the maximum degree of G.In this paper we prove this conjecture for planar graphs with girth at least 5 and for planar graphs not containing cycles of length 4,6,8 and 9. We also show that if G is planar with girth at least 6. Moreover, we find an upper bound for the acyclic chromatic index of planar graphs without cycles of length 4. Namely, we prove that if G is such a graph, then .     

On maximum planar induced subgraphs

The nonplanar vertex deletion or vertex deletion vd(G) of a graph G is the smallest nonnegative integer k, such that the removal of k vertices from G produces a planar graph G^′. In this case G^′ is said to be a maximum planar induced subgraph of G. We solve a problem proposed by Yannakakis: find the threshold for the maximum degree of a graph G such that, given a graph G and a nonnegative integer k, to decide whether vd(G)?k is NP-complete. We prove that it is NP-complete to decide whether a maximum degree 3 graph G and a nonnegative integer k satisfy vd(G)?k. We prove that unless P=NP there is no polynomial-time approximation algorithm with fixed ratio to compute the size of a maximum planar induced subgraph for graphs in general. We prove that it is Max SNP-hard to compute vd(G) when restricted to a cubic input G. Finally, we exhibit a polynomial-time -approximation algorithm for finding a maximum planar induced subgraph of a maximum degree 3 graph.     

Note on robust critical graphs with large odd girth

A graph G is (k+1)-critical if it is not k-colourable but G−e is k-colourable for any edge e∈E(G). In this paper we show that for any integers k≥3 and l≥5 there exists a constant c=c(k,l)>0, such that for all , there exists a (k+1)-critical graph G on n vertices with and odd girth at least ?, which can be made (k−1)-colourable only by the omission of at least cn² edges.     

Rainbowness of cubic plane graphs

The rainbowness, rb(G), of a connected plane graph G is the minimum number k such that any colouring of vertices of the graph G using at least k colours involves a face all vertices of which receive distinct colours. For a connected cubic plane graph G we prove that

     

On the Hadwiger's conjecture for graph products

The Hadwiger number η(G) of a graph G is the largest integer h such that the complete graph on h nodes K_h is a minor of G. Equivalently, η(G) is the largest integer such that any graph on at most η(G) nodes is a minor of G. The Hadwiger's conjecture states that for any graph G, η(G)?χ(G), where χ(G) is the chromatic number of G. It is well-known that for any connected undirected graph G, there exists a unique prime factorization with respect to Cartesian graph products. If the unique prime factorization of G is given as G₁□G₂□?□G_k, where each G_i is prime, then we say that the product dimension of G is k. Such a factorization can be computed efficiently.In this paper, we study the Hadwiger's conjecture for graphs in terms of their prime factorization. We show that the Hadwiger's conjecture is true for a graph G if the product dimension of G is at least . In fact, it is enough for G to have a connected graph M as a minor whose product dimension is at least , for G to satisfy the Hadwiger's conjecture. We show also that if a graph G is isomorphic to F^d for some F, then η(G)?χ(G)^{⌊(d-1)/2⌋}, and thus G satisfies the Hadwiger's conjecture when d?3. For sufficiently large d, our lower bound is exponentially higher than what is implied by the Hadwiger's conjecture.Our approach also yields (almost) sharp lower bounds for the Hadwiger number of well-known graph products like d-dimensional hypercubes, Hamming graphs and the d-dimensional grids. In particular, we show that for the d-dimensional hypercube H_d, . We also derive similar bounds for G^d for almost all G with n nodes and at least edges.     

Distance irredundance and connected domination numbers of a graph

Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number γ_k(G), the connected k-domination number ; the k-independent domination number and the k-irredundance number ir_k(G). The authors prove that if an ir_k-set X is a k-independent set of G, then , and that for k?2, if ir_k(G)=1, if ir_k(G) is odd, and if ir_k(G) is even, which generalize some known results.     

Super s-restricted edge-connectivity of vertex-transitive graphs

Let G be a connected graph with vertex-set V(G)and edge-set E(G).A subset F of E(G)is an s-restricted edge-cut of G if G-F is disconnected and every component of G-F has at least s vertices.Letλs(G)be the minimum size of all s-restricted edge-cuts of G andξs(G)=min{|[X,V(G)\X]|:|X|=s,G[X]is connected},where[X,V(G)\X]is the set of edges with exactly one end in X.A graph G with an s-restricted edge-cut is called super s-restricted edge-connected,in short super-λs,ifλs(G)=ξs(G)and every minimum s-restricted edge-cut of G isolates one component G[X]with|X|=s.It is proved in this paper that a connected vertex-transitive graph G with degree k5 and girth g5 is super-λs for any positive integer s with s 2g or s 10 if k=g=6.