A construction of chromatic index critical graphs

We prove that if K is an undirected, simple, connected graph of even order which is of class one, regular of degree p ≥ 2 and such that the subgraph induced by any three vertices is either connected or null, then any graph G obtained from K by splitting any vertex is p-critical. We find that various constructions of critical graphs by S. Fiorini are special cases of a corollary of this result.     

On cubic non‐Cayley vertex‐transitive graphs

In 1983, the second author [D. Maru?i?, Ars Combinatoria 16B (1983), 297–302] asked for which positive integers n there exists a non‐Cayley vertex‐transitive graph on n vertices. (The term non‐Cayley numbers has later been given to such integers.) Motivated by this problem, Feng [Discrete Math 248 (2002), 265–269] asked to determine the smallest valency ?(n) among valencies of non‐Cayley vertex‐transitive graphs of order n. As cycles are clearly Cayley graphs, ?(n)?3 for any non‐Cayley number n. In this paper a goal is set to determine those non‐Cayley numbers n for which ?(n) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non‐Cayley vertex‐transitive graphs of order n. It is known that for a prime p every vertex‐transitive graph of order p, p² or p³ is a Cayley graph, and that, with the exception of the Coxeter graph, every cubic non‐Cayley vertex‐transitive graph of order 2p, 4p or 2p² is a generalized Petersen graph. In this paper the next natural step is taken by proving that every cubic non‐Cayley vertex‐transitive graph of order 4p², p>7 a prime, is a generalized Petersen graph. In addition, cubic non‐Cayley vertex‐transitive graphs of order 2p^k, where p>7 is a prime and k?p, are characterized. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 77–95, 2012     

Über symmetrische Graphen vom grad fünf

Let G be a connected 1-transitive graph of valency five. It is shown that the order of a vertex stabilizer divides 5 · 3² · 2¹⁷. A theorem of A. Gardiner bounding the order of a vertex stabilizer of a 2-transitive graph of valency 1 + p,p prime, is reproved.     

Matching Properties in Total Domination Vertex Critical Graphs

A vertex subset S of a graph G = (V,E) is a total dominating set if every vertex of G is adjacent to some vertex in S. The total domination number of G, denoted by γ _t (G), is the minimum cardinality of a total dominating set of G. A graph G with no isolated vertex is said to be total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, γ _t (G?v) < γ _t (G). A total domination vertex critical graph G is called k-γ _t -critical if γ _t (G) = k. In this paper we first show that every 3-γ _t -critical graph G of even order has a perfect matching if it is K _1,5-free. Secondly, we show that every 3-γ _t -critical graph G of odd order is factor-critical if it is K _1,5-free. Finally, we show that G has a perfect matching if G is a K _1,4-free 4-γ _t (G)-critical graph of even order and G is factor-critical if G is a K _1,4-free 4-γ _t (G)-critical graph of odd order.     

Remarks on the critical graph conjecture

The vertex-critical graph conjecture (critical graph conjecture respectively) states that every vertex-critical (critical) graph has an odd number of vertices. In this note we prove that if G is a critical graph of even order, then G has at least three vertices of less-than-maximum valency. In addition, if G is a 3-critical multigraph of smallest even order, then G is simple and has no triangles.     

Even graphs

A nontrivial connected graph G is called even if for each vertex v of G there is a unique vertex v such that d(v, v ) = diam G. Special classes of even graphs are defined and compared to each other. In particular, an even graph G is called symmetric if d(u, v) + d(u, v ) = diam G for all u, v ∈ V(G). Several properties of even and symmetric even graphs are stated. For an even graph of order n and diameter d other than an even cycle it is shown that n ≥ 3d – 1 and conjectured that n ≥ 4d – 4. This conjecture is proved for symmetric even graphs and it is shown that for each pair of integers n, d with n even, d ≥ 2 and n ≥ 4d – 4 there exists an even graph of order n and diameter d. Several ways of constructing new even graphs from known ones are presented.     

Medians of arbitrary graphs

For each vertex u in a connected graph H, the distance of u is the sum of the distances from u to each of the vertices v of H. A vertex of minimum distance in H is called a median vertex. It is shown that for any graph G there exists a graph H for which the subgraph of H induced by the median vertices is isomorphic to G.     

Tetravalent vertex‐transitive graphs of order 4p

A graph is vertex‐transitive if its automorphism group acts transitively on vertices of the graph. A vertex‐transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this article, the tetravalent vertex‐transitive non‐Cayley graphs of order 4p are classified for each prime p. As a result, there are one sporadic and five infinite families of such graphs, of which the sporadic one has order 20, and one infinite family exists for every prime p>3, two families exist if and only if p≡1 (mod 8) and the other two families exist if and only if p≡1 (mod 4). For each family there is a unique graph for a given order. © 2011 Wiley Periodicals, Inc.     

On double and multiple interval graphs

In this paper we discuss a generalization of the familiar concept of an interval graph that arises naturally in scheduling and allocation problems. We define the interval number of a graph G to be the smallest positive integer t for which there exists a function f which assigns to each vertex u of G a subset f(u) of the real line so that f(u) is the union of t closed intervals of the real line, and distinct vertices u and v in G are adjacent if and only if f(u) and f(v)meet. We show that (1) the interval number of a tree is at most two, and (2) the complete bipartite graph K_{m, n} has interval number ?(mn + 1)/(m + n)?.     

Construction of Antimagic Labeling for the Cartesian Product of Regular Graphs

An antimagic labeling of a graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1, 2, . . . , q} such that all vertex weights are pairwise distinct, where a vertex weight is the sum of labels of all edges incident with the vertex. A graph is antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that that every connected graph, except K ₂, is antimagic. Recently, using completely separating systems, Phanalasy et al. showed that for each k 3 2, q 3 \binomk+12{k\geq 2,\,q\geq\binom{k+1}{2}} with k|2q, there exists an antimagic k-regular graph with q edges and p = 2q/k vertices. In this paper we prove constructively that certain families of Cartesian products of regular graphs are antimagic.     

Semisymmetric cubic graphs of order 16<Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>

An undirected graph without isolated vertices is said to be semisymmetric if its full automorphism group acts transitively on its edge set but not on its vertex set. In this paper, we inquire the existence of connected semisymmetric cubic graphs of order 16p ². It is shown that for every odd prime p, there exists a semisymmetric cubic graph of order 16p ² and its structure is explicitly specified by giving the corresponding voltage rules generating the covering projections.     

Bounds on a generalized domination parameter

Abstract

If n is an integer, n ≥ 2 and u and v are vertices of a graph G, then u and v are said to be K_n-adjacent vertices of G if there is a subgraph of G, isomorphic to K_n , containing u and v. For n ≥ 2, a K_n- dominating set of G is a set D of vertices such that every vertex of G belongs to D or is K_n-adjacent to a vertex of D. The K_n-domination number γK_n (G) of G is the minimum cardinality among the K_n-dominating sets of vertices of G. It is shown that, for n ε {3,4}, if G is a graph of order p with no K_n-isolated vertex, then γK_n (G) ≤ p/n. We establish that this is a best possible upper bound. It is shown that the result is not true for n ≥ 5.     

3‐Factor‐Criticality of Vertex‐Transitive Graphs

A graph of order n is p ‐factor‐critical, where p is an integer of the same parity as n, if the removal of any set of p vertices results in a graph with a perfect matching. 1‐factor‐critical graphs and 2‐factor‐critical graphs are factor‐critical graphs and bicritical graphs, respectively. It is well known that every connected vertex‐transitive graph of odd order is factor‐critical and every connected nonbipartite vertex‐transitive graph of even order is bicritical. In this article, we show that a simple connected vertex‐transitive graph of odd order at least five is 3‐factor‐critical if and only if it is not a cycle.     

Results on the edge-connectivity of graphs

It is shown that if G is a graph of order p ≥ 2 such that deg u + deg v ≥ p ? 1 for all pairs u, v of nonadjacent vertices, then the edge-connectivity of G equals the minimum degree of G. Furthermore, if deg u + deg v ≥ p for all pairs u, v of nonadjacent vertices, then either p is even and G is isomorphic to $\text{K}{}_{\mathrm{<mtext>p</mtext><mtext>2</mtext>}}\text{× K}{}_2$ or every minimum cutset of edges of G consists of the collection of edges incident with a vertex of least degree.     

Critical Facets of the Stable Set Polytope

Dedicated to the memory of Paul Erdős A facet of the stable set polytope of a graph G can be viewed as a generalization of the notion of an -critical graph. We extend several results from the theory of -critical graphs to facets. The defect of a nontrivial, full-dimensional facet of the stable set polytope of a graph G is defined by . We prove the upper bound for the degree of any node u in a critical facet-graph, and show that can occur only when . We also give a simple proof of the characterization of critical facet-graphs with defect 2 proved by Sewell [11]. As an application of these techniques we sharpen a result of Surányi [13] by showing that if an -critical graph has defect and contains nodes of degree , then the graph is an odd subdivision of . Received October 23, 1998     

Perfect Matchings in Total Domination Critical Graphs

A graph is total domination edge-critical if the addition of any edge decreases the total domination number, while a graph with minimum degree at least two is total domination vertex-critical if the removal of any vertex decreases the total domination number. A 3 _t EC graph is a total domination edge-critical graph with total domination number 3 and a 3 _t VC graph is a total domination vertex-critical graph with total domination number 3. A graph G is factor-critical if G − v has a perfect matching for every vertex v in G. In this paper, we show that every 3 _t EC graph of even order has a perfect matching, while every 3 _t EC graph of odd order with no cut-vertex is factor-critical. We also show that every 3 _t VC graph of even order that is K _1,7-free has a perfect matching, while every 3 _t VC graph of odd order that is K _1,6-free is factor-critical. We show that these results are tight in the sense that there exist 3 _t VC graphs of even order with no perfect matching that are K _1,8-free and 3 _t VC graphs of odd order that are K _1,7-free but not factor-critical.     

On total domination vertex critical graphs of high connectivity

A graph is called γ-critical if the removal of any vertex from the graph decreases the domination number, while a graph with no isolated vertex is γ_t-critical if the removal of any vertex that is not adjacent to a vertex of degree 1 decreases the total domination number. A γ_t-critical graph that has total domination number k, is called k-γ_t-critical. In this paper, we introduce a class of k-γ_t-critical graphs of high connectivity for each integer k≥3. In particular, we provide a partial answer to the question “Which graphs are γ-critical and γ_t-critical or one but not the other?” posed in a recent work [W. Goddard, T.W. Haynes, M.A. Henning, L.C. van der Merwe, The diameter of total domination vertex critical graphs, Discrete Math. 286 (2004) 255-261].     

Paired Domination Vertex Critical Graphs

Let γ _pr (G) denote the paired domination number of graph G. A graph G with no isolated vertex is paired domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, γ _pr (G – v) < γ _pr (G). We call these graphs γ _pr -critical. In this paper, we present a method of constructing γ _pr -critical graphs from smaller ones. Moreover, we show that the diameter of a γ _pr -critical graph is at most and the upper bound is sharp, which answers a question proposed by Henning and Mynhardt [The diameter of paired-domination vertex critical graphs, Czechoslovak Math. J., to appear]. Xinmin Hou: Research supported by NNSF of China (No.10701068 and No.10671191).     

Vertex-transitive graphs: Symmetric graphs of prime valency

Let G be a group acting symmetrically on a graph Σ, let G₁ be a subgroup of G minimal among those that act symmetrically on Σ, and let G₂ be a subgroup of G₁ maximal among those normal subgroups of G₁ which contain no member except 1 which fixes a vertex of Σ. The most precise result of this paper is that if Σ has prime valency p, then either Σ is a bipartite graph or G₂ acts regularly on Σ or G₁ | G₂ is a simple group which acts symmetrically on a graph of valency p which can be constructed from Σ and does not have more vertices than Σ. The results on vertex-transitive groups necessary to establish results like this are also included.     

Note on a conjecture of toft

A conjecture of Toft [17] asserts that any 4-critical graph (or equivalently, every 4-chromatic graph) contains a fully odd subdivision ofK ₄. We show that if a graphG has a degree three nodev such thatG-v is 3-colourable, then eitherG is 3-colourable or it contains a fully oddK ₄. This resolves Toft's conjecture in the special case where a 4-critical graph has a degree three node, which is in turn used to prove the conjecture for line-graphs. The proof is constructive and yields a polynomial algorithm which given a 3-degenerate graph either finds a 3-colouring or exhibits a subgraph that is a fully odd subdivision ofK ₄. (A graph is 3-degenerate if every subgraph has some node of degree at most three.)