Circular Chromatic Indices of Regular Graphs

The circular chromatic index of a graph G, written , is the minimum r permitting a function such that whenever e and are adjacent. It is known that for any , there is a 3‐regular simple graph G with . This article proves the following results: Assume is an odd integer. For any , there is an n‐regular simple graph G with . For any , there is an n‐regular multigraph G with .     

Toughness and Vertex Degrees

We study theorems giving sufficient conditions on the vertex degrees of a graph G to guarantee G is t‐tough. We first give a best monotone theorem when , but then show that for any integer , a best monotone theorem for requires at least nonredundant conditions, where grows superpolynomially as . When , we give an additional, simple theorem for G to be t‐tough, in terms of its vertex degrees.     

A Proof of a Conjecture of Ohba

We prove a conjecture of Ohba that says that every graph G on at most vertices satisfies .     

Partial Online List Coloring of Graphs

For a graph G, let be the maximum number of vertices of G that can be colored whenever each vertex of G is given t permissible colors. Albertson, Grossman, and Haas conjectured that if G is s‐choosable and , then . In this article, we consider the online version of this conjecture. Let be the maximum number of vertices of G that can be colored online whenever each vertex of G is given t permissible colors online. An analog of the above conjecture is the following: if G is online s‐choosable and then . This article generalizes some results concerning partial list coloring to online partial list coloring. We prove that for any positive integers , . As a consequence, if s is a multiple of t, then . We also prove that if G is online s‐choosable and , then and for any , .     

Average Degree in Graph Powers

The kth power of a simple graph G, denoted by , is the graph with vertex set where two vertices are adjacent if they are within distance k in G. We are interested in finding lower bounds on the average degree of . Here we prove that if G is connected with minimum degree and , then G⁴ has average degree at least . We also prove that if G is a connected d‐regular graph on n vertices with diameter at least , then the average degree of is at least Both these results are shown to be essentially best possible; the second is best possible even when is arbitrarily large.     

Cycle‐Saturated Graphs with Minimum Number of Edges

A graph G is called H‐saturated if it does not contain any copy of H, but for any edge e in the complement of G, the graph contains some H. The minimum size of an n‐vertex H‐saturated graph is denoted by . We prove holds for all , where is a cycle with length k. A graph G is H‐semisaturated if contains more copies of H than G does for . Let be the minimum size of an n‐vertex H‐semisaturated graph. We have We conjecture that our constructions are optimal for . © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 203–215, 2013     

A Combined Logarithmic Bound on the Chromatic Index of Multigraphs

For a multigraph G, the integer round‐up of the fractional chromatic index provides a good general lower bound for the chromatic index . For an upper bound, Kahn 1996 showed that for any real there exists a positive integer N so that whenever . We show that for any multigraph G with order n and at least one edge, ). This gives the following natural generalization of Kahn's result: for any positive reals , there exists a positive integer N so that + c whenever . We also compare the upper bound found here to other leading upper bounds.     

Independent Domination in Cubic Graphs

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number of G, denoted by , is the minimum cardinality of an independent dominating set. In this article, we show that if is a connected cubic graph of order n that does not have a subgraph isomorphic to K_{2, 3}, then . As a consequence of our main result, we deduce Reed's important result [Combin Probab Comput 5 (1996), 277–295] that if G is a cubic graph of order n, then , where denotes the domination number of G.     

Partitioning a Graph into Highly Connected Subgraphs

Given , a k‐proper partition of a graph G is a partition of such that each part P of induces a k‐connected subgraph of G. We prove that if G is a graph of order n such that , then G has a 2‐proper partition with at most parts. The bounds on the number of parts and the minimum degree are both best possible. We then prove that if G is a graph of order n with minimum degree where , then G has a k‐proper partition into at most parts. This improves a result of Ferrara et al. ( Discrete Math 313 (2013), 760–764), and both the degree condition and the number of parts is best possible up to the constant c.     

Total Vertex Irregularity Strength of Dense Graphs

Consider a graph of minimum degree δ and order n. Its total vertex irregularity strength is the smallest integer k for which one can find a weighting such that for every pair of vertices of G. We prove that the total vertex irregularity strength of graphs with is bounded from above by . One of the cornerstones of the proof is a random ordering of the vertices generated by order statistics.     

Destroying Noncomplete Regular Components in Graph Partitions

We prove that if G is a graph and such that then can be partitioned into sets such that and contains no noncomplete ‐regular components for each . In particular, the vertex set of any graph G can be partitioned into sets, each of which induces a disjoint union of triangles and paths.     

List Colorings with Distinct List Sizes,the Case of Complete Bipartite Graphs

Let be a function on the vertex set of the graph . The graph G is f‐choosable if for every collection of lists with list sizes specified by f there is a proper coloring using colors from the lists. The sum choice number, , is the minimum of , over all functions f such that G is f‐choosable. It is known (Alon, Surveys in Combinatorics, 1993 (Keele), London Mathematical Society Lecture Note Series, Vol. 187, Cambridge University Press, Cambridge, 1993, pp. 1–33, Random Struct Algor 16 (2000), 364–368) that if G has average degree d, then the usual choice number is at least , so they grow simultaneously. In this article, we show that can be bounded while the minimum degree . Our main tool is to give tight estimates for the sum choice number of the unbalanced complete bipartite graph .     

The Maximum Number of Dominating Induced Matchings

A matching M of a graph G is a dominating induced matching (DIM) of G if every edge of G is either in M or adjacent with exactly one edge in M. We prove sharp upper bounds on the number of DIMs of a graph G and characterize all extremal graphs. Our results imply that if G is a graph of order n, then ; provided G is triangle‐free; and provided and G is connected.     

A Hypergraph Version of a Graph Packing Theorem by Bollobás and Eldridge

Two n‐vertex hypergraphs G and H pack, if there is a bijection such that for every edge , the set is not an edge in H. Extending a theorem by Bollobás and Eldridge on graph packing to hypergraphs, we show that if and n‐vertex hypergraphs G and H with with no edges of size 0, 1, and n do not pack, then either

1. one of G and H contains a spanning graph‐star, and each vertex of the other is contained in a graph edge, or
2. one of G and H has edges of size not containing a given vertex, and for every vertex x of the other hypergraph some edge of size does not contain x.

     

Counterexamples to the List Square Coloring Conjecture

The square G² of a graph G is the graph defined on such that two vertices u and v are adjacent in G² if the distance between u and v in G is at most 2. Let and be the chromatic number and the list chromatic number of a graph H, respectively. A graph H is called chromatic‐choosable if . It is an interesting problem to find graphs that are chromatic‐choosable. Kostochka and Woodall (Choosability conjectures and multicircuits, Discrete Math., 240 (2001), 123–143) conjectured that for every graph G, which is called List Square Coloring Conjecture. In this article, we give infinitely many counter examples to the conjecture. Moreover, we show that the value can be arbitrarily large.     

Gallai's Conjecture For Graphs of Girth at Least Four

In 1966, Gallai conjectured that for any simple, connected graph G having n vertices, there is a path‐decomposition of G having at most paths. In this article, we show that for any simple graph G having girth , there is a path‐decomposition of G having at most paths, where is the number of vertices of odd degree in G and is the number of nonisolated vertices of even degree in G.     

Maximizing H‐Colorings of a Regular Graph

For graphs G and H, a homomorphism from G to H, or H‐coloring of G, is an adjacency preserving map from the vertex set of G to the vertex set of H. Our concern in this article is the maximum number of H‐colorings admitted by an n‐vertex, d‐regular graph, for each H. Specifically, writing for the number of H‐colorings admitted by G, we conjecture that for any simple finite graph H (perhaps with loops) and any simple finite n‐vertex, d‐regular, loopless graph G, we have where is the complete bipartite graph with d vertices in each partition class, and is the complete graph on vertices.Results of Zhao confirm this conjecture for some choices of H for which the maximum is achieved by . Here, we exhibit for the first time infinitely many nontrivial triples for which the conjecture is true and for which the maximum is achieved by .We also give sharp estimates for and in terms of some structural parameters of H. This allows us to characterize those H for which is eventually (for all sufficiently large d) larger than and those for which it is eventually smaller, and to show that this dichotomy covers all nontrivial H. Our estimates also allow us to obtain asymptotic evidence for the conjecture in the following form. For fixed H, for all d‐regular G, we have where as . More precise results are obtained in some special cases.     

The Merrifield–Simmons Conjecture Holds for Bipartite Graphs

Let be a graph and the number of independent (vertex) sets of G. Then the Merrifield–Simmons conjecture states that the sign of the term only depends on the parity of the distance of the vertices in G. We prove that the conjecture holds for bipartite graphs by considering a generalization of the term, where vertex subsets instead of vertices are deleted.     

Improper Coloring of Sparse Graphs with a Given Girth,II: Constructions

A graph G is ‐colorable if can be partitioned into two sets and so that the maximum degree of is at most j and of is at most k. While the problem of verifying whether a graph is (0, 0)‐colorable is easy, the similar problem with in place of (0, 0) is NP‐complete for all nonnegative j and k with . Let denote the supremum of all x such that for some constant every graph G with girth g and for every is ‐colorable. It was proved recently that . In a companion paper, we find the exact value . In this article, we show that increasing g from 5 further on does not increase much. Our constructions show that for every g, . We also find exact values of for all g and all .