We determine the values of s and t for which there is a coloring of the edges of the complete bip... more We determine the values of s and t for which there is a coloring of the edges of the complete bipartite graph Ks,t which admits only the identity automorphism. In particular, we show that for a given s with a few small exceptions, there is such a coloring with c colors if and only r ≤ t ≤ cs − r where r = blogc(s − 1)c + 1 if s ≤ c or if s ≥ c + 1 and s ≥ c1+blogc(s−1)c − blogcblogc(s− 1)cc and that r = blogc(s− 1)c+ 2 if s ≤ c1+blogc(s−1)c − blogcblogc(s− 1)cc − 2. When s = c1+blogc(s−1)c − blogcblogc(s− 1)cc−1 then r will be one of these two values and we can determine which recursively. Harary and Jacobson [1] examined the minimum number of edges that need to be oriented so that the resulting mixed graph has the trivial automorphism group and determined some values of s and t for which this number exists for the complete bipartite graph Ks,t. These are values for which there is a mixed graph resulting from orienting some of the edges with only the trivial automorphism. Such an or...
We examine the size $s(n)$ of a smallest tournament having the arcs of an acyclic tournament on $... more We examine the size $s(n)$ of a smallest tournament having the arcs of an acyclic tournament on $n$ vertices as a minimum feedback arc set. Using an integer linear programming formulation we obtain lower bounds $s(n) \geq 3n - 2 - \lfloor \log_2 n \rfloor$ or $s(n) \geq 3n - 1 - \lfloor \log_2 n \rfloor$, depending on the binary expansion of $n$. When $n = 2^k - 2^t$ we show that the bounds are tight with $s(n) = 3n - 2 - \lfloor \log_2 n \rfloor$. One view of this problem is that if the 'teams' in a tournament are ranked to minimize inconsistencies there is some tournament with $s(n)$ 'teams' in which $n$ are ranked wrong. We will also pose some questions about conditions on feedback arc sets, motivated by our proofs, which ensure equality between the maximum number of arc disjoint cycles and the minimum size of a feedback arc set in a tournament.
A pebbling move refers to the act of removing two pebbles from one vertex and placing one pebble ... more A pebbling move refers to the act of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The goal of graph pebbling is: Given an initial distribution of pebbles, use pebbling moves to reach a specified goal vertex called the root . The pebbling number of a graph $$\pi (G)$$ π ( G ) is the minimum number of pebbles needed so every distribution of $$\pi (G)$$ π ( G ) pebbles can reach every choice of the root. We introduce a new variant of graph pebbling, a game between two players. One player aims to move a pebble to the root and the other player aims to prevent this. We show configurations of various classes of graphs for which each player has a winning strategy. We will characterize the winning player for a specific class of diameter two graphs.
We will consider two easily stated combinatorial problems, determining if a given sequence of int... more We will consider two easily stated combinatorial problems, determining if a given sequence of integers could arise as the numbers of wins in a round robin tournament and determining if a diagram of relations can be realized as ‘comes before’ for a set of intervals in time. Both problems will be given equivalent formulations of determining if a system of inequalities has a solution. These particular linear systems relate to circulation and distance problems in digraphs. Theorems for the digraph problems will yield proofs of the combinatorial problems. Necessary and sufficient conditions for the digraph problems will be derived directly from Farkas’ Lemma. Farkas’ lemma is a theorem of the alternative for systems of linear inequalities, stating that either such a system has a solution or it has a simple certificate of inconsistency.
Two basic exercises in graph theory are characterizing degree sequences of trees and degree seque... more Two basic exercises in graph theory are characterizing degree sequences of trees and degree sequences of multigraphs. For each there are several proofs using different approaches. The goal here is to examine the interesting exercises that arise when we combine these two problems by looking at degree sequences of multigraphs with an underlying forest like structure.
We examine degree characterizations for 2-multitrees. These are multigraphs with underlying tree ... more We examine degree characterizations for 2-multitrees. These are multigraphs with underlying tree structure and at most 2 copies of each edge. We provide characterizations for both when a degree bipartition is given and when it is not given.
A graph is a probe interval graph if its vertices can be partitioned into probes and nonprobes wi... more A graph is a probe interval graph if its vertices can be partitioned into probes and nonprobes with an interval associated to each vertex so that vertices are adjacent if and only if their corresponding intervals intersect and at least one of them is a probe. A graph G = (V, E) is a tolerance graph if each vertex v is an element of V can be associated to an interval I(v) of the real line and a positive real number t(v) such that uv is an element of E if and only if vertical bar I(u) boolean AND I(v)vertical bar >= min \t(u), t(v)\. In this paper we present O(vertical bar V vertical bar + vertical bar E vertical bar) recognition algorithms for both bipartite probe interval graphs and bipartite tolerance graphs. We also give a new structural characterization for each class which follows from the algorithms.
Given a c-edge-colored graph G on n vertices, we define the degree matrix M as the c× n matrix wh... more Given a c-edge-colored graph G on n vertices, we define the degree matrix M as the c× n matrix whose entry dij is the degree of color i at vertex vj . We show that the obvious necessary conditions for a c× n matrix to be the degree matrix of a c-edge-colored forest on n vertices are also sufficient. It is well known that non-negative integers d1, d2, . . . , dn form a degree list of a forest on n vertices if and only if ∑ di is even and ∑ di ≤ 2s − 2 where s is the number of non-zero entries in d1, d2, . . . , dn. We are interested in similar conditions for edge colored forests where we specify the number of incident edges of each color. Given an edge colored forest and any set of colors, the edges of those colors induce a forest. Thus the degree sums for these colors must satisfy the conditions for uncolored forests. We will show that this necessary condition is also sufficient. The three color version of our problem is related to results in [1] and [2]. In those papers, two of the...
The graph Ramsey number R(G, H) is the smallest integer n such that every 2-coloring of the edges... more The graph Ramsey number R(G, H) is the smallest integer n such that every 2-coloring of the edges of Kn contains either a red copy of G or a blue copy of H . We find the largest star that can be removed from Kn such that the underlying graph is still forced to have a red G or a blue H . Thus, we introduce the star-avoiding Ramsey number r∗(G, H) as the smallest integer k such that every 2-coloring of the edges of Kn − K1,n−1−k contains either a red copy of G or a blue copy of H . We find the star-avoiding Ramsey number for trees versus complete graphs, multiple copies of K2 and K3, and paths versus a 4-cycle. In addition to finding the star-avoiding Ramsey numbers, the critical graphs are classified for R(Tn, Km), R(nK2, mK2) and R(Pn, C4).
A chain packing H in a graph is a subgraph satisfying given degree constraints at the vertices. I... more A chain packing H in a graph is a subgraph satisfying given degree constraints at the vertices. Its size is the number of odd degree vertices in the subgraph. An odd subtree packing is a chain packing which is a forest in which all non-isolated vertices have odd degree in the forest. We show that for a given graph and degree constraints, the size of a maximum chain packing and a maximum odd subtree packing are the same but the same does not hold for a version in which the sum of given weights on the odd degree vertices is to be maximized. We also note a reduction to weighted capacitated b-matching for finding a maximum size chain packing, maximum size odd subtree packing and maximum weight chain packing. The main result of this note is the proof that a min-max formula generalizing the Berge-Tutte formula for matching holds for chain packing.
A d-dimensional Perfect Factor is a collection of periodic arrays in which every k-ary (n1×· · ·×... more A d-dimensional Perfect Factor is a collection of periodic arrays in which every k-ary (n1×· · ·×nd) matrix appears appears exactly once (periodically). The one dimensional case, with a collection of size one, is known as a De Bruijn cycle. The 1and 2-dimensional versions have proven highly applicable in areas such as coding, communications, and location sensing. Here we focus on results in higher dimensions for factors with each ni = 2.
Ag raph isf-choosable if for every collection of lists with list sizes specied by f there is a pr... more Ag raph isf-choosable if for every collection of lists with list sizes specied by f there is a proper coloring using colors from the lists. The sum choice number is the minimum over all choosable functions f of the sum of the sizes in f .W e show that the sum choice number of a 2 n array (equivalent to list edge coloring K2;n and to list vertex coloring the cartesian product K22Kn )i sn 2 + d5n=3e.
We determine the values of s and t for which there is a coloring of the edges of the complete bip... more We determine the values of s and t for which there is a coloring of the edges of the complete bipartite graph Ks,t which admits only the identity automorphism. In particular, we show that for a given s with a few small exceptions, there is such a coloring with c colors if and only r ≤ t ≤ cs − r where r = blogc(s − 1)c + 1 if s ≤ c or if s ≥ c + 1 and s ≥ c1+blogc(s−1)c − blogcblogc(s− 1)cc and that r = blogc(s− 1)c+ 2 if s ≤ c1+blogc(s−1)c − blogcblogc(s− 1)cc − 2. When s = c1+blogc(s−1)c − blogcblogc(s− 1)cc−1 then r will be one of these two values and we can determine which recursively. Harary and Jacobson [1] examined the minimum number of edges that need to be oriented so that the resulting mixed graph has the trivial automorphism group and determined some values of s and t for which this number exists for the complete bipartite graph Ks,t. These are values for which there is a mixed graph resulting from orienting some of the edges with only the trivial automorphism. Such an or...
We examine the size $s(n)$ of a smallest tournament having the arcs of an acyclic tournament on $... more We examine the size $s(n)$ of a smallest tournament having the arcs of an acyclic tournament on $n$ vertices as a minimum feedback arc set. Using an integer linear programming formulation we obtain lower bounds $s(n) \geq 3n - 2 - \lfloor \log_2 n \rfloor$ or $s(n) \geq 3n - 1 - \lfloor \log_2 n \rfloor$, depending on the binary expansion of $n$. When $n = 2^k - 2^t$ we show that the bounds are tight with $s(n) = 3n - 2 - \lfloor \log_2 n \rfloor$. One view of this problem is that if the 'teams' in a tournament are ranked to minimize inconsistencies there is some tournament with $s(n)$ 'teams' in which $n$ are ranked wrong. We will also pose some questions about conditions on feedback arc sets, motivated by our proofs, which ensure equality between the maximum number of arc disjoint cycles and the minimum size of a feedback arc set in a tournament.
A pebbling move refers to the act of removing two pebbles from one vertex and placing one pebble ... more A pebbling move refers to the act of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The goal of graph pebbling is: Given an initial distribution of pebbles, use pebbling moves to reach a specified goal vertex called the root . The pebbling number of a graph $$\pi (G)$$ π ( G ) is the minimum number of pebbles needed so every distribution of $$\pi (G)$$ π ( G ) pebbles can reach every choice of the root. We introduce a new variant of graph pebbling, a game between two players. One player aims to move a pebble to the root and the other player aims to prevent this. We show configurations of various classes of graphs for which each player has a winning strategy. We will characterize the winning player for a specific class of diameter two graphs.
We will consider two easily stated combinatorial problems, determining if a given sequence of int... more We will consider two easily stated combinatorial problems, determining if a given sequence of integers could arise as the numbers of wins in a round robin tournament and determining if a diagram of relations can be realized as ‘comes before’ for a set of intervals in time. Both problems will be given equivalent formulations of determining if a system of inequalities has a solution. These particular linear systems relate to circulation and distance problems in digraphs. Theorems for the digraph problems will yield proofs of the combinatorial problems. Necessary and sufficient conditions for the digraph problems will be derived directly from Farkas’ Lemma. Farkas’ lemma is a theorem of the alternative for systems of linear inequalities, stating that either such a system has a solution or it has a simple certificate of inconsistency.
Two basic exercises in graph theory are characterizing degree sequences of trees and degree seque... more Two basic exercises in graph theory are characterizing degree sequences of trees and degree sequences of multigraphs. For each there are several proofs using different approaches. The goal here is to examine the interesting exercises that arise when we combine these two problems by looking at degree sequences of multigraphs with an underlying forest like structure.
We examine degree characterizations for 2-multitrees. These are multigraphs with underlying tree ... more We examine degree characterizations for 2-multitrees. These are multigraphs with underlying tree structure and at most 2 copies of each edge. We provide characterizations for both when a degree bipartition is given and when it is not given.
A graph is a probe interval graph if its vertices can be partitioned into probes and nonprobes wi... more A graph is a probe interval graph if its vertices can be partitioned into probes and nonprobes with an interval associated to each vertex so that vertices are adjacent if and only if their corresponding intervals intersect and at least one of them is a probe. A graph G = (V, E) is a tolerance graph if each vertex v is an element of V can be associated to an interval I(v) of the real line and a positive real number t(v) such that uv is an element of E if and only if vertical bar I(u) boolean AND I(v)vertical bar >= min \t(u), t(v)\. In this paper we present O(vertical bar V vertical bar + vertical bar E vertical bar) recognition algorithms for both bipartite probe interval graphs and bipartite tolerance graphs. We also give a new structural characterization for each class which follows from the algorithms.
Given a c-edge-colored graph G on n vertices, we define the degree matrix M as the c× n matrix wh... more Given a c-edge-colored graph G on n vertices, we define the degree matrix M as the c× n matrix whose entry dij is the degree of color i at vertex vj . We show that the obvious necessary conditions for a c× n matrix to be the degree matrix of a c-edge-colored forest on n vertices are also sufficient. It is well known that non-negative integers d1, d2, . . . , dn form a degree list of a forest on n vertices if and only if ∑ di is even and ∑ di ≤ 2s − 2 where s is the number of non-zero entries in d1, d2, . . . , dn. We are interested in similar conditions for edge colored forests where we specify the number of incident edges of each color. Given an edge colored forest and any set of colors, the edges of those colors induce a forest. Thus the degree sums for these colors must satisfy the conditions for uncolored forests. We will show that this necessary condition is also sufficient. The three color version of our problem is related to results in [1] and [2]. In those papers, two of the...
The graph Ramsey number R(G, H) is the smallest integer n such that every 2-coloring of the edges... more The graph Ramsey number R(G, H) is the smallest integer n such that every 2-coloring of the edges of Kn contains either a red copy of G or a blue copy of H . We find the largest star that can be removed from Kn such that the underlying graph is still forced to have a red G or a blue H . Thus, we introduce the star-avoiding Ramsey number r∗(G, H) as the smallest integer k such that every 2-coloring of the edges of Kn − K1,n−1−k contains either a red copy of G or a blue copy of H . We find the star-avoiding Ramsey number for trees versus complete graphs, multiple copies of K2 and K3, and paths versus a 4-cycle. In addition to finding the star-avoiding Ramsey numbers, the critical graphs are classified for R(Tn, Km), R(nK2, mK2) and R(Pn, C4).
A chain packing H in a graph is a subgraph satisfying given degree constraints at the vertices. I... more A chain packing H in a graph is a subgraph satisfying given degree constraints at the vertices. Its size is the number of odd degree vertices in the subgraph. An odd subtree packing is a chain packing which is a forest in which all non-isolated vertices have odd degree in the forest. We show that for a given graph and degree constraints, the size of a maximum chain packing and a maximum odd subtree packing are the same but the same does not hold for a version in which the sum of given weights on the odd degree vertices is to be maximized. We also note a reduction to weighted capacitated b-matching for finding a maximum size chain packing, maximum size odd subtree packing and maximum weight chain packing. The main result of this note is the proof that a min-max formula generalizing the Berge-Tutte formula for matching holds for chain packing.
A d-dimensional Perfect Factor is a collection of periodic arrays in which every k-ary (n1×· · ·×... more A d-dimensional Perfect Factor is a collection of periodic arrays in which every k-ary (n1×· · ·×nd) matrix appears appears exactly once (periodically). The one dimensional case, with a collection of size one, is known as a De Bruijn cycle. The 1and 2-dimensional versions have proven highly applicable in areas such as coding, communications, and location sensing. Here we focus on results in higher dimensions for factors with each ni = 2.
Ag raph isf-choosable if for every collection of lists with list sizes specied by f there is a pr... more Ag raph isf-choosable if for every collection of lists with list sizes specied by f there is a proper coloring using colors from the lists. The sum choice number is the minimum over all choosable functions f of the sum of the sizes in f .W e show that the sum choice number of a 2 n array (equivalent to list edge coloring K2;n and to list vertex coloring the cartesian product K22Kn )i sn 2 + d5n=3e.
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